Exact solutions are obtained for a model of the longitudinal displacement along an elastic tapered bar due to a force applied at its blunt end. A formula for velocity amplification is given; it specifies the velocity of the pointed end of the bar shortly after it feels the influence of the force. For a bar with an exponentially decreasing cross-sectional area, the velocity is magnified by twice an exponential function of length. This result has applications in the design of piezoelectric drills. In addition, we discuss the differences between the motions of rigid and elastic bars during the transient before one complete reflection of the wave induced by a force applied to an end of the bar. In this regime, force is proportional to velocity for elastic bars with constant cross-sectional areas. While the force–velocity relationship is more complicated for tapered elastic bars, their exact relationship is determined.
Issue Section:
Technical Papers
1.
Huang
, T.
, Kimura
, Y.
, and Yanagimachi
, R.
, 1996, “The Use of Piezo Micromanipulation for Intracytoplasmic Sperm Injection of Human Oocytes
,” J. Assist Reprod. Genet.
1058-0468, 13
(4
), pp. 320
–328
.2.
Katayose
, H.
, Yanagida
, K.
, Shinoki
, T.
, Kawahara
, T.
, Horiuchi
, T.
, and Sato
, A.
, 1999, “Efficient Injection of Bull Spermatozoa into Oocytes Using a Piezo-driven Pipette
,” Theriogenology
0093-691X, 52
, pp. 1215
–1224
.3.
Kimura
, Y.
, and Yanagimachi
, R.
, 1995, “Intracytoplasmic Sperm Injection in the Mouse
,” Biol. Reprod.
0006-3363, 52
(4
), pp. 709
–720
.4.
Nakayama
, T.
, Fijiwara
, H.
, Tastumi
, K.
, Fujita
, K.
, Higuchi
, T.
, and Mori
, T.
, 1998, “A New Assisted Hatching Technique Using a Piezo-Micromanipulator
,” Fertil. Steril.
0015-0282, 69
(4
), pp. 784
–788
.5.
Takeuchi
, S.
, Minoura
, H.
, Shibahara
, T.
, Shen
, X.
, Futamura
, N.
, and Toyoda
, N.
, 2001, “Comparison of Piezo-Assisted Micromanipulation with Conventional Micromanipulation for Intracytoplasmic Sperm Injection into Human Oocytes
,” Gynecol. Obstet. Invest.
0378-7346, 52
, pp. 158
–162
.6.
Dunstan
, D. J.
, and Spain
, I. L.
, 1989, “Technology of Diamond Anvil High-Pressure Cells: I. Principles, Design and Construction
,” J. Phys. E
0022-3735, 22
, pp. 913
–923
.7.
ASTM Standard.
ASTM Designation: G32-92. Standard Test Method for Cavitation Erosion Using Vibratory Apparatus
, Philadelphia
, PA.8.
Lal
, A.
, and White
, R. M.
, 1995, “Silicon Microfabricated Horns for Power Ultrasonics
”, Proceedings Transducers 95
, Stockholm
, Sweden, July 25–29.9.
Lal
, A.
, and White
, R. M.
, 1995, “Micromachined Silicon Needle for Ultrasonic Surgery
,” Proceedings IEEE Ultrasonics Symposium
, Seattle, WA, November 7–10, IEEE, Piscataway, NJ, pp. 1593
–1596
.10.
Wendt
, O.
, Siewert
, C.
, Luth
, T.
, Felix
, R.
, and Boenick
, U.
, 2001, “Cutting Speeds and Success of Biopsy with Different Punch Biopsy Instruments
,” Radiologe
0033-832X, 41
(6
), pp. 484
–490
in (German).11.
Eisner
, E.
, 1967, “Complete Solutions of the “Webster” Horn Equation
,” J. Acoust. Soc. Am.
0001-4966, 41
(4
), pp. 1126
–1967
.12.
Mason
, W. P.
, and Wick
, R. F.
, 1951, “A Barium Titanate Transducer Capable of Large Motion at an Ultrasonic Frequency
,” J. Acoust. Soc. Am.
0001-4966, 23
, pp. 209
–214
.13.
Eisner
, E.
, 1963, “Design of Sonic Amplitude Transformers for High Magnification
,” J. Acoust. Soc. Am.
0001-4966, 35
(9
), pp. 1367
–1377
.14.
Graff
, K. F.
, 1991, Wave Motion in Elastic Solids
, Reprint
, Dover, New York.15.
Kolsky
, H.
, 1953, Stress Waves in Solids
, Clarendon Press
, Oxford, U.K.16.
Redwood
, M.
, 1960, Mechanical Waveguides
, Pergamon
, Bristol, U.K.17.
McLachlan
, N. W.
, 1958, Ordinary Non-linear Differential Equations in Engineering and Physical Sciences
, 2nd ed., Clarendon
, Oxford, U.K.18.
Zhou
, G.
, and Li
, M.
, 2000, “A Study on Ultrasonic Solid Horns for Flexural Mode
,” J. Acoust. Soc. Am.
0001-4966, 107
(3
), pp. 1358
–1362
.19.
Sherrit
, S.
, et al., 2004, “Novel Horn Designs for Power Ultrasonics
,” Proceedings of the IEEE International Ultrasonics Symposium
, Montreal, August 24
–27
.20.
Lynnworth
, L. C.
, 1989, Ultrasonic Measurements for Process Control: Theory, Techniques, Applications
, Academic
, New York.21.
Ffowcs Williams
, J. E.
, and Hawkings
, D. L.
, 1969, 1969, “Sound Generated by Turbulence and Surfaces in Arbitrary Motion
,” Philos. Trans. R. Soc. London, Ser. A
0962-8428, 264
, pp. 321
–342
.22.
Rienstra
, S. W.
, and Hirschberg
, A.
, 2001, “An Introduction to Acoustics
,” Rep. No. IWDE 01-03, Eindhoven University of Technology, Eindhoven
, The Netherlands.23.
Courant
, R.
, and Hilbert
, D.
, 1962, Methods of Mathematical Physics
, Interscience
, New York.24.
Al Kaufman
, A.
, Levshin
, A. L.
, and Larner
, K.
, 2004, “The Role of Wave Propagation in the Motion of an Elastic Body
,” Eur. J. Phys.
0143-0807, 25
, pp. 257
–268
.25.
Baldock
, G. R.
, and Bridgeman
, T.
, 1981, The Mathematical Theory of Wave Motion
, Ellis Horwood Limited
, Chichester, UK.26.
John
, F.
, 1982, Partial Differential Equations
, 4th ed., Springer
, New York.Copyright © 2007
by American Society of Mechanical Engineers
You do not currently have access to this content.
