Ultrasonic Butt Welding of Aluminum, Aluminum Alloy
Content
Ultrasonics 40 (2002) 371–374 www.elsevier.com/locate/ultras
Ultrasonic butt welding of aluminum, aluminum alloy and stainless steel plate specimens
Jiromaru Tsujino *, Kazuaki Hidai, Atsushi Hasegawa, Ryoichi Kanai, Hisanori Matsuura, Kaoru Matsushima, Tetsugi Ueoka
Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama 221-8686, Japan
Abstract Welding characteristics of aluminum, aluminum alloy and stainless steel plate specimens of 6.0 mm thickness by a 15 kHz ultrasonic butt welding system were studied. There are no detailed welding condition data of these specimens although the joining of these materials are required due to anticorrosive and high strength characteristics for not only large specimens but small electronic parts especially. These specimens of 6.0 mm thickness were welded end to end using a 15 kHz ultrasonic butt welding equipment with a vibration source using eight bolt-clamped Langevin type PZT transducers and a 50 kW static induction thyristor power amplifier. The stainless steel plate specimens electrolytically polished were joined with welding strength almost equal to the material strength under rather large vibration amplitude of 25 lm (peak-to-zero value), static pressure 70 MPa and welding time of 1.0–3.0 s. The hardness of stainless steel specimen adjacent to a welding surface increased about 20% by ultrasonic vibration. Ó 2002 Elsevier Science B.V. All rights reserved.
Keywords: Ultrasonic welding; Ultrasonic butt welding; Welding of aluminum and stainless steel; Welding of stainless steel specimens; Hardness distribution of welded part
1. Introduction Welding characteristics of stainless steel and aluminum plate specimens of 6.0 mm thickness by a 15 kHz ultrasonic butt welding system were studied. There are no detailed welding condition data of these specimens although the joining of these materials are required due to anticorrosive and high strength characteristics for not only large specimens but small electronic parts especially. Aluminum and stainless steel specimens, and stainless steel specimens of 6.0 mm thickness were welded end to end using a 15 kHz ultrasonic butt welding equipment with a vibration source using eight bolt-clamped Langevin type PZT transducers of 70 mm diameter and a 50 kW static induction thyristor power amplifier. The aluminum, aluminum alloy and stainless steel plate specimens were joined with almost equal to aluminum
* Corresponding author. Tel.: +81-45-481-5661; fax: +81-45-4917915. E-mail addresses: [email protected], [email protected] (J. Tsujino).
strength. The stainless steel plate specimens electrolytically polished were joined with welding strength almost equal to the material strength under rather large vibration amplitude of 25 lm (peak-to-zero value), static pressure 70 MPa and welding time of 1.0–3.0 s. From the results of measurements of hardness adjacent to the welded part, hardness of aluminum was found to decrease near the welded surface. Hardness of stainless steel decreases at the areas approximately 2 mm away from the welded surface but increases by approximately 20% near the welded surface. The increase in hardness due to ultrasonic vibration was found in welding of these materials, which was on contrary to the phenomenon of decreasing hardness due to ultrasonic vibration observed in many industrial applications. The combinations of aluminum–stainless steel and stainless–stainless steel specimens were reported as that the materials are possible to join each other by ultrasonic welding, but there were no report about welding conditions of large specimens with welded strength near to the material strength excepting small specimens. Using the butt welding equipment, as an example, the obtained required vibration velocity is real value without
0041-624X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 1 - 6 2 4 X ( 0 2 ) 0 0 1 2 4 - 5
372
J. Tsujino et al. / Ultrasonics 40 (2002) 371–374
slippage such as a conventional ultrasonic lapped welding equipment. The obtained data in this paper are fundamental welding conditions for joining ring or spot shape parts of the products such as small and thin motors of hard disk drive and the others. Of course, adequate welding equipments must be developed for special applications including complex vibration systems.
2. Configuration of a butt welding equipment Fig. 1 shows an arrangement of an upper and a lower welding tip, welding specimens and clamping vices for ultrasonic butt welding of specimens end to end. The ultrasonic butt welding system consists of a 15 kHz powerful ultrasonic vibration source, an upper passive vibration system and a welding frame with hydraulic static pressure sources to clamp the welding specimens. The vibration source was driven by a 50 kW static induction thyristor power amplifier [1–4]. The driving side specimen is clamped by a metal vice for avoiding the fatigue failures by transverse vibration of the specimen at a transverse nodal position where was carefully chosen for the material and dimensions of the welding specimens. The clamping position is shown in Fig. 1. The welded surfaces were driven to vibrate parallel to each other by welding tips and were welded together end to end. The welding specimens used were 6-mmthick and 10–20-mm-wide aluminum (JISA1100P, JISA5052P) and stainless steel (SUS304P) plates. The welding surfaces were finished to be flat by a milling machine. The specimens were degreased and cleaned by chlorothene and given no further treatment.
Fig. 2. Welded conditions of 6-mm-thick, 10-mm-wide stainless steel plate specimens. (1) Upper view, (2) side view and (3) cross-section of welded 6-mm-thick, 10-mm-wide (a) aluminum alloy–stainless steel and (b) stainless steel plate specimens.
3. Welding characteristics Fig. 2 shows (1) upper view, (2) side views and (3) cross-sections of the welded (a) aluminum–stainless and (b) stainless steel–stainless steel specimens. The welded surface of aluminum–stainless steel is linear, and thin
burrs are produced only at the side of the aluminum specimens because the hardness of aluminum is less than that of stainless steel. The welded surface are difficult to observe in the cross-section of the completely joined stainless steel specimens but some cracks are found in defectively joined specimens. Fig. 3 shows broken conditions of welded aluminum alloy-stainless steel and stainless steel specimens after tensile test. The relationship between welding tip vibration amplitude and weld strength of 6-mm-thick and 20-mmwide aluminum and stainless steel specimens is shown in Fig. 4. The welding tip vibration amplitude is altered from 17 to 27 lm (peak-to-zero value). The static clamping pressure at the welding surface is 20 MPa and the welding time is maintained at 2.0 s. The weld strengths obtained are near to the aluminum specimen’s strength over all vibration amplitude range. Electric input power was approximately 4 kW.
Fig. 1. 15 kHz ultrasonic butt welding equipment using an upper and a lower welding tips, and hydraulic vices for clamping welding specimens and inducing static pressure to welding surface.
Fig. 3. Broken conditions of welded specimens after tensile strength tests. (a) Aluminum alloy–stainless steel specimen and (b) stainless steel specimen.
J. Tsujino et al. / Ultrasonics 40 (2002) 371–374
373
Fig. 4. Relationship between vibration amplitude, input power and weld strength of 6.0-mm-thick and 20-mm-wide pure aluminum and electrolytically polished stainless steel plate specimens.
The relationship between welding tip vibration amplitude and weld strength of 6-mm-thick and 10-mmwide stainless steel specimens is shown in Fig. 5. The welding tip vibration amplitude is altered from 18 to 30 lm (peak-to-zero value). The static clamping pressure at the welding surface is 70 MPa and the welding time is maintained at 3.0 s. The maximum weld strength about 450 MPa is obtained under vibration amplitude 23 lm. Hardness distributions are measured by a Vickers micro-hardness tester along the cross-sections of the welded specimen at the upper, center and lower parts across a welding surface. Fig. 6 shows the hardness distribution along a weldment section across a welded surface of 6-mm-thick aluminum alloy and stainless steel specimens. A decrease in hardness of the aluminum specimen adjacent to the welded surface is noted, while that on the stainless steel side is observed in the area 2 mm away from the welded surface and an increase in hardness is noted adjacent to the welded surface. The increment is largest along the central part of the section as compared with the upper and lower parts, which are close to a clamping vice, due to heat conduction. The
Fig. 6. Hardness distributions along a cross-section of a 6.0-mm-thick and 10-mm-wide welded aluminum alloy–stainless steel specimen at upper, center, lower parts. Welding time: 1.0 s.
maximum increment of hardness measured is over 20% in the stainless steel specimen. Fig. 7 shows the hardness distribution of 6-mm-thick stainless steel specimens. The decrease and increase in hardness are also noted adjacent to the welded surface of the stainless steel specimen. The increase in hardness is observed for the first time, which is contrary to the decreases in hardness being noted in various high-power ultrasonic applications. Fig. 8 shows the relationship between the temperature, the heating time and the hardness of the stainless steel specimen, which was measured after heating in the furnace. The hardness of the stainless steel increases according to rather low heating temperature and short time due to the aging effect of stainless steel. The hardness of the stainless steel specimen increases slightly as temperature increases to 950 °C, decreases in
Fig. 5. Relationship between vibration amplitude, input power and weld strength of 6.0-mm-thick and 10-mm-wide stainless steel plate specimens.
Fig. 7. Hardness distributions along a cross-section of a 6.0-mm-thick and 10-mm-wide welded stainless steel specimen at upper, center, lower parts. Welding time: 2.5 s.
374
J. Tsujino et al. / Ultrasonics 40 (2002) 371–374
Fig. 8. Relationship heating time and hardness of 6-mm-thick stainless steel specimen under temperatures of 350, 400 and 500 °C.
the range between 950 and 1000 °C, and then increases when the temperature exceeds 1200 °C [5]. In the absence of ultrasonic vibration and static pressure, the increase in hardness of stainless steel specimens could be attributed to such high temperature rise of over 1200 °C at the weldment. However the hardness increase effect should actually ascribed mainly to the vibration as no visible light emission due to the high temperature rise at the welded part was observed. In fact, the maximum temperature rise directly measured from the thermo-electromotive force among aluminum, copper and steel was only 441 °C [4].
4. Conclusion Welding characteristics of stainless steel and aluminum plate specimens of 6.0 mm thickness by a 15 kHz ultrasonic butt welding system were studied. Aluminum and stainless steel specimens, and stainless steel specimens of 6.0 mm thickness were welded end to end using a 15 kHz ultrasonic butt welding equipment with a vibration source using eight bolt-clamped Langevin type PZT transducers of 70 mm diameter and a 50 kW static induction thyristor power amplifier. The aluminum, aluminum alloy and stainless steel plate specimens were joined with almost equal to aluminum strength. The stainless steel plate specimens electrolytically polished were joined with welding strength almost
equal to the material strength under rather large vibration amplitude of 25 lm (peak-to-zero value), static pressure 70 MPa and welding time of 1.0–3.0 s. Hardness of aluminum was found to decrease near the welded surface. Hardness of stainless steel decreases at the areas approximately 2 mm away from the welded surface but increases by approximately 20% near the welded surface. The phenomenon of increasing hardness due to ultrasonic vibration was observed although only decreasing hardness was noted in previous studies. The hardness increasing effect may be applied to the partial hardening of materials due to vibration; in contrast, the hardness decreasing effect may be applied to the relaxation of residual stress at the welded metal surface. The combinations of aluminum–stainless steel and stainless–stainless steel specimens were reported as that the materials are possible to join each other by ultrasonic welding, but there were no report about welding conditions of large specimens with welded strength near to the material strength excepting small specimens as yet. Using the ultrasonic butt welding equipment, as an example, the required vibration velocities revealed in this paper are real values without slippage between the welding tip and the upper specimen in the case of a conventional ultrasonic lapped welding equipment. The obtained data are fundamental welding conditions for joining ring or spot shape parts of the products such as small and thin motors of hard disk drive and the others. Of course, adequate welding equipments must be developed for special applications including complex vibration systems. References
[1] J. Tsujino, T. Ueoka, Ultrasonic butt welding of aluminum, anticorrosive aluminum and copper plate specimens, in: Proc. IEEE 1988 Ultrasonics Symp., IEEE, New York, 1989, p. 496. [2] J. Tsujino, T. Ueoka, Y. Susaki, M. Ogawa, Y. Hirasawa, Y. Fujita, Jpn. J. Appl. Phys. 32 (1993) 2441. [3] J. Tsujino, T. Ueoka, I. Watanabe, Y. Kimura, T. Mori, K. Hasegawa, Y. Fujita, T. Shiraki, M. Motonaga, New methods of ultrasonic metal welding, in: Proc. IEEE 1993 Ultrasonics Symp., IEEE, New York, 1994, p. 405. [4] J. Tsujino, T. Ueoka, Y. Asada, S. Taniguchi, Y. Iwamura, Jpn. J. Appl. Phys. 37 (1998) 2996. [5] J. Tsujino, T. Ueoka, T. Kashino, F. Sugahara, Jpn. J. Appl. Phys. 38 (1999) 4254.
Ultrasonic butt welding of aluminum, aluminum alloy and stainless steel plate specimens
Jiromaru Tsujino *, Kazuaki Hidai, Atsushi Hasegawa, Ryoichi Kanai, Hisanori Matsuura, Kaoru Matsushima, Tetsugi Ueoka
Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama 221-8686, Japan
Abstract Welding characteristics of aluminum, aluminum alloy and stainless steel plate specimens of 6.0 mm thickness by a 15 kHz ultrasonic butt welding system were studied. There are no detailed welding condition data of these specimens although the joining of these materials are required due to anticorrosive and high strength characteristics for not only large specimens but small electronic parts especially. These specimens of 6.0 mm thickness were welded end to end using a 15 kHz ultrasonic butt welding equipment with a vibration source using eight bolt-clamped Langevin type PZT transducers and a 50 kW static induction thyristor power amplifier. The stainless steel plate specimens electrolytically polished were joined with welding strength almost equal to the material strength under rather large vibration amplitude of 25 lm (peak-to-zero value), static pressure 70 MPa and welding time of 1.0–3.0 s. The hardness of stainless steel specimen adjacent to a welding surface increased about 20% by ultrasonic vibration. Ó 2002 Elsevier Science B.V. All rights reserved.
Keywords: Ultrasonic welding; Ultrasonic butt welding; Welding of aluminum and stainless steel; Welding of stainless steel specimens; Hardness distribution of welded part
1. Introduction Welding characteristics of stainless steel and aluminum plate specimens of 6.0 mm thickness by a 15 kHz ultrasonic butt welding system were studied. There are no detailed welding condition data of these specimens although the joining of these materials are required due to anticorrosive and high strength characteristics for not only large specimens but small electronic parts especially. Aluminum and stainless steel specimens, and stainless steel specimens of 6.0 mm thickness were welded end to end using a 15 kHz ultrasonic butt welding equipment with a vibration source using eight bolt-clamped Langevin type PZT transducers of 70 mm diameter and a 50 kW static induction thyristor power amplifier. The aluminum, aluminum alloy and stainless steel plate specimens were joined with almost equal to aluminum
* Corresponding author. Tel.: +81-45-481-5661; fax: +81-45-4917915. E-mail addresses: [email protected], [email protected] (J. Tsujino).
strength. The stainless steel plate specimens electrolytically polished were joined with welding strength almost equal to the material strength under rather large vibration amplitude of 25 lm (peak-to-zero value), static pressure 70 MPa and welding time of 1.0–3.0 s. From the results of measurements of hardness adjacent to the welded part, hardness of aluminum was found to decrease near the welded surface. Hardness of stainless steel decreases at the areas approximately 2 mm away from the welded surface but increases by approximately 20% near the welded surface. The increase in hardness due to ultrasonic vibration was found in welding of these materials, which was on contrary to the phenomenon of decreasing hardness due to ultrasonic vibration observed in many industrial applications. The combinations of aluminum–stainless steel and stainless–stainless steel specimens were reported as that the materials are possible to join each other by ultrasonic welding, but there were no report about welding conditions of large specimens with welded strength near to the material strength excepting small specimens. Using the butt welding equipment, as an example, the obtained required vibration velocity is real value without
0041-624X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 1 - 6 2 4 X ( 0 2 ) 0 0 1 2 4 - 5
372
J. Tsujino et al. / Ultrasonics 40 (2002) 371–374
slippage such as a conventional ultrasonic lapped welding equipment. The obtained data in this paper are fundamental welding conditions for joining ring or spot shape parts of the products such as small and thin motors of hard disk drive and the others. Of course, adequate welding equipments must be developed for special applications including complex vibration systems.
2. Configuration of a butt welding equipment Fig. 1 shows an arrangement of an upper and a lower welding tip, welding specimens and clamping vices for ultrasonic butt welding of specimens end to end. The ultrasonic butt welding system consists of a 15 kHz powerful ultrasonic vibration source, an upper passive vibration system and a welding frame with hydraulic static pressure sources to clamp the welding specimens. The vibration source was driven by a 50 kW static induction thyristor power amplifier [1–4]. The driving side specimen is clamped by a metal vice for avoiding the fatigue failures by transverse vibration of the specimen at a transverse nodal position where was carefully chosen for the material and dimensions of the welding specimens. The clamping position is shown in Fig. 1. The welded surfaces were driven to vibrate parallel to each other by welding tips and were welded together end to end. The welding specimens used were 6-mmthick and 10–20-mm-wide aluminum (JISA1100P, JISA5052P) and stainless steel (SUS304P) plates. The welding surfaces were finished to be flat by a milling machine. The specimens were degreased and cleaned by chlorothene and given no further treatment.
Fig. 2. Welded conditions of 6-mm-thick, 10-mm-wide stainless steel plate specimens. (1) Upper view, (2) side view and (3) cross-section of welded 6-mm-thick, 10-mm-wide (a) aluminum alloy–stainless steel and (b) stainless steel plate specimens.
3. Welding characteristics Fig. 2 shows (1) upper view, (2) side views and (3) cross-sections of the welded (a) aluminum–stainless and (b) stainless steel–stainless steel specimens. The welded surface of aluminum–stainless steel is linear, and thin
burrs are produced only at the side of the aluminum specimens because the hardness of aluminum is less than that of stainless steel. The welded surface are difficult to observe in the cross-section of the completely joined stainless steel specimens but some cracks are found in defectively joined specimens. Fig. 3 shows broken conditions of welded aluminum alloy-stainless steel and stainless steel specimens after tensile test. The relationship between welding tip vibration amplitude and weld strength of 6-mm-thick and 20-mmwide aluminum and stainless steel specimens is shown in Fig. 4. The welding tip vibration amplitude is altered from 17 to 27 lm (peak-to-zero value). The static clamping pressure at the welding surface is 20 MPa and the welding time is maintained at 2.0 s. The weld strengths obtained are near to the aluminum specimen’s strength over all vibration amplitude range. Electric input power was approximately 4 kW.
Fig. 1. 15 kHz ultrasonic butt welding equipment using an upper and a lower welding tips, and hydraulic vices for clamping welding specimens and inducing static pressure to welding surface.
Fig. 3. Broken conditions of welded specimens after tensile strength tests. (a) Aluminum alloy–stainless steel specimen and (b) stainless steel specimen.
J. Tsujino et al. / Ultrasonics 40 (2002) 371–374
373
Fig. 4. Relationship between vibration amplitude, input power and weld strength of 6.0-mm-thick and 20-mm-wide pure aluminum and electrolytically polished stainless steel plate specimens.
The relationship between welding tip vibration amplitude and weld strength of 6-mm-thick and 10-mmwide stainless steel specimens is shown in Fig. 5. The welding tip vibration amplitude is altered from 18 to 30 lm (peak-to-zero value). The static clamping pressure at the welding surface is 70 MPa and the welding time is maintained at 3.0 s. The maximum weld strength about 450 MPa is obtained under vibration amplitude 23 lm. Hardness distributions are measured by a Vickers micro-hardness tester along the cross-sections of the welded specimen at the upper, center and lower parts across a welding surface. Fig. 6 shows the hardness distribution along a weldment section across a welded surface of 6-mm-thick aluminum alloy and stainless steel specimens. A decrease in hardness of the aluminum specimen adjacent to the welded surface is noted, while that on the stainless steel side is observed in the area 2 mm away from the welded surface and an increase in hardness is noted adjacent to the welded surface. The increment is largest along the central part of the section as compared with the upper and lower parts, which are close to a clamping vice, due to heat conduction. The
Fig. 6. Hardness distributions along a cross-section of a 6.0-mm-thick and 10-mm-wide welded aluminum alloy–stainless steel specimen at upper, center, lower parts. Welding time: 1.0 s.
maximum increment of hardness measured is over 20% in the stainless steel specimen. Fig. 7 shows the hardness distribution of 6-mm-thick stainless steel specimens. The decrease and increase in hardness are also noted adjacent to the welded surface of the stainless steel specimen. The increase in hardness is observed for the first time, which is contrary to the decreases in hardness being noted in various high-power ultrasonic applications. Fig. 8 shows the relationship between the temperature, the heating time and the hardness of the stainless steel specimen, which was measured after heating in the furnace. The hardness of the stainless steel increases according to rather low heating temperature and short time due to the aging effect of stainless steel. The hardness of the stainless steel specimen increases slightly as temperature increases to 950 °C, decreases in
Fig. 5. Relationship between vibration amplitude, input power and weld strength of 6.0-mm-thick and 10-mm-wide stainless steel plate specimens.
Fig. 7. Hardness distributions along a cross-section of a 6.0-mm-thick and 10-mm-wide welded stainless steel specimen at upper, center, lower parts. Welding time: 2.5 s.
374
J. Tsujino et al. / Ultrasonics 40 (2002) 371–374
Fig. 8. Relationship heating time and hardness of 6-mm-thick stainless steel specimen under temperatures of 350, 400 and 500 °C.
the range between 950 and 1000 °C, and then increases when the temperature exceeds 1200 °C [5]. In the absence of ultrasonic vibration and static pressure, the increase in hardness of stainless steel specimens could be attributed to such high temperature rise of over 1200 °C at the weldment. However the hardness increase effect should actually ascribed mainly to the vibration as no visible light emission due to the high temperature rise at the welded part was observed. In fact, the maximum temperature rise directly measured from the thermo-electromotive force among aluminum, copper and steel was only 441 °C [4].
4. Conclusion Welding characteristics of stainless steel and aluminum plate specimens of 6.0 mm thickness by a 15 kHz ultrasonic butt welding system were studied. Aluminum and stainless steel specimens, and stainless steel specimens of 6.0 mm thickness were welded end to end using a 15 kHz ultrasonic butt welding equipment with a vibration source using eight bolt-clamped Langevin type PZT transducers of 70 mm diameter and a 50 kW static induction thyristor power amplifier. The aluminum, aluminum alloy and stainless steel plate specimens were joined with almost equal to aluminum strength. The stainless steel plate specimens electrolytically polished were joined with welding strength almost
equal to the material strength under rather large vibration amplitude of 25 lm (peak-to-zero value), static pressure 70 MPa and welding time of 1.0–3.0 s. Hardness of aluminum was found to decrease near the welded surface. Hardness of stainless steel decreases at the areas approximately 2 mm away from the welded surface but increases by approximately 20% near the welded surface. The phenomenon of increasing hardness due to ultrasonic vibration was observed although only decreasing hardness was noted in previous studies. The hardness increasing effect may be applied to the partial hardening of materials due to vibration; in contrast, the hardness decreasing effect may be applied to the relaxation of residual stress at the welded metal surface. The combinations of aluminum–stainless steel and stainless–stainless steel specimens were reported as that the materials are possible to join each other by ultrasonic welding, but there were no report about welding conditions of large specimens with welded strength near to the material strength excepting small specimens as yet. Using the ultrasonic butt welding equipment, as an example, the required vibration velocities revealed in this paper are real values without slippage between the welding tip and the upper specimen in the case of a conventional ultrasonic lapped welding equipment. The obtained data are fundamental welding conditions for joining ring or spot shape parts of the products such as small and thin motors of hard disk drive and the others. Of course, adequate welding equipments must be developed for special applications including complex vibration systems. References
[1] J. Tsujino, T. Ueoka, Ultrasonic butt welding of aluminum, anticorrosive aluminum and copper plate specimens, in: Proc. IEEE 1988 Ultrasonics Symp., IEEE, New York, 1989, p. 496. [2] J. Tsujino, T. Ueoka, Y. Susaki, M. Ogawa, Y. Hirasawa, Y. Fujita, Jpn. J. Appl. Phys. 32 (1993) 2441. [3] J. Tsujino, T. Ueoka, I. Watanabe, Y. Kimura, T. Mori, K. Hasegawa, Y. Fujita, T. Shiraki, M. Motonaga, New methods of ultrasonic metal welding, in: Proc. IEEE 1993 Ultrasonics Symp., IEEE, New York, 1994, p. 405. [4] J. Tsujino, T. Ueoka, Y. Asada, S. Taniguchi, Y. Iwamura, Jpn. J. Appl. Phys. 37 (1998) 2996. [5] J. Tsujino, T. Ueoka, T. Kashino, F. Sugahara, Jpn. J. Appl. Phys. 38 (1999) 4254.
