3
IN MEMORIAM Dr. Lung-Wen Tsai 1945–2002 Dr. Lung-Wen Tsai, the former technical editor of this journal passed away unexpectedly but peacefully due to a stroke at his home in Riverside, California, on Friday, November 29, 2002. He was a professor of mechanical engineering at the University of California ~UC!, Riverside. His loving wife Lung-Chu, daughter Jule Ann, and son David survive him. Dr. Tsai was born in Taipei, Taiwan, on February 20, 1945, and was the youngest of 10 siblings. A hard worker, Dr. Tsai helped cultivate his family’s farm until he finished his undergraduate studies in mechanical engineering at the National Taiwan University in 1967. He then came to this country and studied mechanical engineering at State University of New York at Buffalo receiving his M.S. degree in 1970. He then went to California and attended Stanford University receiving his Ph.D. in mechanical engineering in 1973. He was the only sibling in his family to leave Taiwan and study in the United States. Before joining the Bourns College of Engineering faculty at UC Riverside in 2000, Dr. Tsai was a professor at the University of Maryland in College Park for over fourteen years where he established world-renowned research and educational programs in robotics and mechanisms, and had been a research engineer with the General Motors Corporation and Hewlett-Packard Company. Dr. Tsai’s research interests were in kinematics, robotics, mechanisms and machine theory, and mechanical design. Dr. Tsai has made several important contributions to these fields starting with his Ph.D. thesis where he developed geometric conditions for studying ‘‘incompletely specified displacements.’’ He then applied this to the design of three-dimensional linkages. The work still remains as a key contribution to kinematics and machine theory. It has also found applications in robotics and trajectory planning. In robotics, he was the first to numerically determine all the real solutions of the most general kinematic equations for six and five degrees of freedom serial robot manipulators. Dr. Tsai has also made several contributions on design of robotic systems which includes design of geared and tendon driven devices and parallel manipulators and platforms. His most recent contribution in this area has been the development of a novel parallel manipulator at UC Riverside. This manipulator is unique in that it uses a parallel articulated and over-constrained kinematic structure but behaves like a conventional X-Y-Z gantry system. Some of his other contributions in mechanical design include his work on design of geared transmission devices, his novel design and analysis of a harmonic balancer, and his work on Oldham Couplings for automotive applications. Dr. Tsai was a Fellow of the American Society of Mechanical Engineers and a member of the Society of Automotive Engineers. He held numerous U.S. patents and was the author of two textbooks, 66 archival journal or transactions papers, and 100 conference and other papers that have been published up to the time of this memorial. His honors include the ASME Melville Medal received in 1985, Best Paper Awards ~1984 ASME Mechanisms Conference, 1989 and 1991 Applied Mechanisms and Robotics Conference!, Arch Colwell Merit Award ~1988 SAE International Congress & Exposition!, South Pointing Chariot Award ~1993 Applied Mechanisms & Robotics Conference!, Presidential Chair Professor at UC Riverside ~2000–2002!, and numerous international invited pro- fessorships and lectureships. In February 2002, Dr. Tsai was elected as an Honorary Professor at National Chiao- Tung University, one of the top five universities in Taiwan. In September 2002, Dr. Tsai was elected as a Fellow of the American Association for the Advancement of Science ~AAAS!, which publishes the peer-reviewed journal Science. Dr. Tsai’s professional activities included serving both as a member and the chair of the mechanisms technical committee within the design division of ASME, general chair of the International Design Engineering Technical Conferences in 2000 and serving as the Technical Editor of the ASME Journal of Mechanical Design, since 1998. Dr. Tsai was a dedicated educator and a scholar, respected by his students and fellow colleagues. He was also devoted to his family, friends, and his profession. He will be missed by those who knew him as well as by the scientific and engineering communities. Following this memorial is a list of Dr. Tsai’s technical publications. Bahram Ravani University of California—Davis Copyright © 2003 by ASME Journal of Mechanical Design SEPTEMBER 2003, Vol. 125 Õ 647 Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 07/27/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use

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Page 1: In Memoriam: Dr. Lung–Wen Tsai - Conference Proceedingsproceedings.asmedigitalcollection.asme.org/data/Journals/JMDEDB/... · IN MEMORIAM Dr. Lung-Wen Tsai 1945–2002 Dr. Lung-Wen

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IN MEMORIAM

Dr. Lung-Wen Tsai1945–2002

Dr. Lung-Wen Tsai, the former technical editor of this journal passed away unexpectedly but peacefully dstroke at his home in Riverside, California, on Friday, November 29, 2002. He was a professor of mecengineering at the University of California~UC!, Riverside. His loving wife Lung-Chu, daughter Jule Ann, and sDavid survive him.

Dr. Tsai was born in Taipei, Taiwan, on February 20, 1945, and was the youngest of 10 siblings. A hard wDr. Tsai helped cultivate his family’s farm until he finished his undergraduate studies in mechanical engineethe National Taiwan University in 1967. He then came to this country and studied mechanical engineeringUniversity of New York at Buffalo receiving his M.S. degree in 1970. He then went to California and atteStanford University receiving his Ph.D. in mechanical engineering in 1973. He was the only sibling in his famleave Taiwan and study in the United States.

Before joining the Bourns College of Engineering faculty at UC Riverside in 2000, Dr. Tsai was a professorUniversity of Maryland in College Park for over fourteen years where he established world-renowned reseaeducational programs in robotics and mechanisms, and had been a research engineer with the GeneraCorporation and Hewlett-Packard Company.

Dr. Tsai’s research interests were in kinematics, robotics, mechanisms and machine theory, and medesign. Dr. Tsai has made several important contributions to these fields starting with his Ph.D. thesis wdeveloped geometric conditions for studying ‘‘incompletely specified displacements.’’ He then applied thisdesign of three-dimensional linkages. The work still remains as a key contribution to kinematics and mtheory. It has also found applications in robotics and trajectory planning.

In robotics, he was the first to numerically determine all the real solutions of the most general kinematic eqfor six and five degrees of freedom serial robot manipulators. Dr. Tsai has also made several contributions oof robotic systems which includes design of geared and tendon driven devices and parallel manipulatplatforms. His most recent contribution in this area has been the development of a novel parallel manipulatoRiverside. This manipulator is unique in that it uses a parallel articulated and over-constrained kinematic sbut behaves like a conventional X-Y-Z gantry system. Some of his other contributions in mechanical designhis work on design of geared transmission devices, his novel design and analysis of a harmonic balancerwork on Oldham Couplings for automotive applications.

Dr. Tsai was a Fellow of the American Society of Mechanical Engineers and a member of the SociAutomotive Engineers. He held numerous U.S. patents and was the author of two textbooks, 66 archival jotransactions papers, and 100 conference and other papers that have been published up to the time of this mHis honors include the ASME Melville Medal received in 1985, Best Paper Awards~1984 ASME MechanismsConference, 1989 and 1991 Applied Mechanisms and Robotics Conference!, Arch Colwell Merit Award~1988 SAEInternational Congress & Exposition!, South Pointing Chariot Award~1993 Applied Mechanisms & RoboticConference!, Presidential Chair Professor at UC Riverside~2000–2002!, and numerous international invited professorships and lectureships. In February 2002, Dr. Tsai was elected as an Honorary Professor at NationaTung University, one of the top five universities in Taiwan. In September 2002, Dr. Tsai was elected as a Fethe American Association for the Advancement of Science~AAAS!, which publishes the peer-reviewed journScience.

Dr. Tsai’s professional activities included serving both as a member and the chair of the mechanisms tecommittee within the design division of ASME, general chair of the International Design Engineering TecConferences in 2000 and serving as the Technical Editor of the ASMEJournal of Mechanical Design, since 1998.

Dr. Tsai was a dedicated educator and a scholar, respected by his students and fellow colleagues. Hedevoted to his family, friends, and his profession. He will be missed by those who knew him as well asscientific and engineering communities. Following this memorial is a list of Dr. Tsai’s technical publications

Bahram RavaniUniversity of California—Davis

Copyright © 2003 by ASMEof Mechanical Design SEPTEMBER 2003, Vol. 125 Õ 647

://proceedings.asmedigitalcollection.asme.org/ on 07/27/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Page 2: In Memoriam: Dr. Lung–Wen Tsai - Conference Proceedingsproceedings.asmedigitalcollection.asme.org/data/Journals/JMDEDB/... · IN MEMORIAM Dr. Lung-Wen Tsai 1945–2002 Dr. Lung-Wen

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Books

Tsai, L. W., 2000,Mechanism Design: Enumeration of Kinematic Structures According to Function, CRC Press,Boca Raton, FL.

Tsai, L. W., 1999,Robot Analysis: The Mechanics of Serial and Parallel Manipulators, John Wiley & Sons, NewYork, N.Y.

Papers in Archival Journals and Transactions1 Fang, Y., and Tsai, L. W., 2003, ‘‘Inverse Velocity and Singularity Analysis of Low-DOF Serial Manipulators,’’ Journal of Robotic Sys

Vol. 20, No. 4, pp. 177–188.2 Fang, Y., and Tsai, L. W., 2003, ‘‘Feasible Motion Solutions for Serial Manipulators at Singular Configurations,’’ ASME Journ

Mechanical Design, Vol. 125, No. 1, pp. 61–69.3 Kim, H. S., and Tsai, L. W., 2003, ‘‘Design Optimization of a Cartesian Parallel Manipulator,’’ ASME Journal of Mechanical Design

125, No. 1, pp. 43–51.4 Joshi, S., and Tsai, L. W., 2003, ‘‘The Kinematics of a Class of 3-DOF, 4-Legged Parallel Manipulators,’’ ASME Journal of Mech

Design, Vol. 125, No. 1, pp. 52–60.5 Joshi, S., and Tsai, L. W., 2003, ‘‘A Comparison Study of Two 3-DOF Parallel Manipulators: One with Three and the Other Wit

Supporting Legs,’’ IEEE Transactions on Robotics and Automation, Vol. 19, No. 2, pp. 200–209.6 Kim, H. S., and Tsai, L. W., 2003, ‘‘Kinematic Synthesis of a Spatial 3-RPS Parallel Manipulator,’’ ASME Journal of Mechanical D

Vol. 125, No. 1, pp. 92–97.7 Fang, Y., and Tsai, L. W., 2002, ‘‘Structure Synthesis of a Class of 4-DOF and 5-DOF Parallel Manipulators With Identical Limb Struc

Journal of Robotic Systems, Vol. 21, No. 9, pp. 799–810.8 Joshi, S., and Tsai, L. W., 2002, ‘‘Jacobian Analysis of Limited-DOF Parallel Manipulators,’’ ASME Journal of Mechanical Design, Vo

No. 2, pp. 254–258.9 Tsai, L. W., and Joshi, S., 2002, ‘‘Kinematic Analysis of 3-DOF Position Mechanisms for Use in Hybrid Kinematic Machines,’’ A

Journal of Mechanical Design, Vol. 124, No. 2, pp. 245–253.10 Lee, J. J., and Tsai, L. W., 2002, ‘‘Structural Synthesis of Multi-Fingered Hands,’’ ASME Journal of Mechanical Design, Vol. 124, No

272–276.11 Kimbal, C., and Tsai, L. W., 2002, ‘‘Modeling of Flexural Sections Subjected to Arbitrary End Loads,’’ASME Journal of Mechanical D

Vol. 124, No. 2, pp. 223–235.12 Schultz, G. A., Tsai, L. W., Higuchi, N., and Tong, I. C., 2001, ‘‘Development of a Novel Parallel Hybrid Transmission,’’ SAE Transac

Journal of Passenger Cars—Mechanical Systems, paper No. 2001-01-0875; also in SAE Special Publications on Transmission &Systems, SP-1598.

13 Tsai, L. W., Schultz, G., and Higuchi, N., 2001, ‘‘A Novel Parallel Hybrid Transmission,’’ ASME Journal of Mechanical Design, Vol.No. 2, pp. 161–168.

14 Tsai, L. W., and Joshi, S., 2000, ‘‘Kinematics and Optimization of a Spatial 3-UPU Parallel Manipulator,’’ ASME Journal of MechDesign, Vol. 122, No. 4, pp. 439–446.

15 Tsai, L. W., 2000, ‘‘Solving the Inverse Dynamics of a Stewart-Gough Manipulator by the Principle of Virtual Work,’’ ASME JournMechanical Design, Vol. 122, No. 1, pp. 3–9.

16 Hsieh, H. I., and Tsai, L. W., 1998, ‘‘The Selection of a Most Efficient Clutching Sequence Associated With an Automatic TransmMechanism,’’ ASME Journal of Mechanical Design, Vol. 120, No. 4, pp. 514–519.

17 Tsai, L. W., Chen, D. Z., and Lin, T. W., 1998, ‘‘Dynamic Analysis of Geared Robotic Mechanisms Using Graph Theory,’’ ASME JouMechanical Design, Vol. 120, No. 2, pp. 240–244.

18 Shieh, W. B., Tsai, L. W., Azarm, S., and Tits, A., 1998, ‘‘A New Class of Six-Bar Mechanisms with Symmetrical Coupler Curves,’’ AJournal of Mechanical Design, Vol. 120, No. 1, pp. 150–153.

19 Shieh, W. B., Tsai, L. W., and Azarm, S., 1997, ‘‘Design and Optimization of a One-DOF Six-Bar Leg Mechanism for a Walking MacJournal of Robotic Systems, Vol. 14, No. 12, pp. 871–880.

20 Hsieh, H. I., and Tsai, L. W., 1997, ‘‘A Methodology for Enumeration of Clutching Sequences Associated with Epicyclic-Type AutoTransmission Mechanisms,’’ SAE Transactions, Journal of Passenger Cars, Sec. 6, Vol. 105, pp. 928–936.

21 Ou, Y. J., and Tsai, L. W., 1996, ‘‘Isotropic Design of Tendon-Driven Manipulators,’’ ASME Journal of Mechanical Design, Vol. 118,pp. 360–366.

22 Chatterjee, G., and Tsai, L. W., 1996, ‘‘Computer Aided Sketching of Epicyclic-Type Automatic Transmission Gear Trains,’’ ASME Jof Mechanical Design, Vol. 118, No. 3, pp. 405–411.

23 Hsieh, H. I., and Tsai, L. W., 1996, ‘‘Kinematic Analysis of Epicyclic-Type Transmission Mechanisms Using the Concept of FundaGeared Entities,’’ ASME Journal of Mechanical Design, Vol. 118, No. 2, pp. 294–299.

24 Shieh, W. B., Tsai, L. W., Azarm, S., and Tits, A. L., 1996, ‘‘Multiobjective Optimization of a Leg Mechanism With Various SpConfigurations for Force Reduction,’’ ASME Journal of Mechanical Design, Vol. 118, No. 2, pp. 179–185.

25 Tahmasebi, F., and Tsai, L. W., 1995, ‘‘On the Stiffness of a Novel Six-DOF Parallel Mini-manipulator,’’ Journal of Robotic System12, No. 12, pp. 845–856.

26 Tsai, L. W., 1995, ‘‘Design of Tendon-Driven Manipulators,’’ ASME Transactions, A Special Combined Issue of the ASME JourMechanical Design and Journal of Vibration and Acoustics, Vol. 117~B!, pp. 80–86.

27 Lee, J. J., and Tsai, L. W., 1995, ‘‘Dynamic Simulation of Tendon-Driven Manipulators,’’ Journal of Applied Mechanisms and Robotic2, No. 1, pp. 40–60.

28 Chatterjee, G., and Tsai, L. W., 1995, ‘‘Enumeration of Epicyclic-Type Automatic Transmission Gear Trains,’’ SAE Transactions, JoPassenger Cars, Sec. 6, Vol. 103, pp. 1415–1426.

29 Tsai, L. W., and Chang, S. L., 1994, ‘‘Backlash Control via Redundant Drives: An Experimental Verification,’’ ASME Journal of MechDesign, Vol. 116, No. 3, pp. 963–967.

30 Tahmasebi, F., and Tsai, L. W., 1994, ‘‘Closed-Form Direct Kinematics Solution of a New Parallel Mini-Manipulator,’’ ASME JourMechanical Design, Vol. 116, No. 4, pp. 1141–1147.

31 Huang, F., Chen, R. J., and Tsai, L. W., 1994, ‘‘The Use of Random Steer Test Data for Vehicle Parameter Estimation,’’ SAE TranVol. 102, pp. 1329–1340.

32 Chen, D. Z., and Tsai, L. W., 1994, ‘‘Synthesis of Geared Robotic Mechanisms with Maximum Acceleration Capacity,’’ Journal of AMechanisms and Robotics, Vol. 1, No. 3, pp. 17–25.

33 Tahmasebi, F., and Tsai, L. W., 1994, ‘‘Workspace and Singularity Analysis of a Novel Six-DOF Parallel Mini-Manipulator,’’ JourApplied Mechanisms and Robotics, Vol. 1, No. 2, pp. 31–40.

34 Ou, Y. J., and Tsai, L. W., 1993, ‘‘Kinematic Synthesis of Tendon-Driven Manipulators With Isotropic Transmission Characteristics,’’Journal of Mechanical Design, Vol. 115, No. 4, pp. 884–891.

35 Lee, J. J., and Tsai, L. W., 1993, ‘‘Torque Resolver Design for Tendon-Driven Manipulators,’’ASME Journal of Mechanical Design, VoNo. 4, pp. 877–883.

36 Tsai, L. W., and Tahmasebi, F., 1993, ‘‘Synthesis and Analysis of a New Class of Six-DOF Parallel Mini-Manipulators,’’ Journal of RSystems, Vol. 10, No. 5, pp. 561–580.

37 Chen, D. Z., and Tsai, L. W., 1993, ‘‘Kinematic and Dynamic Synthesis of Geared Robotic Mechanisms,’’ ASME Transactions, JoMechanical Design, Vol. 115, No. 2, pp. 241–246.

38 Chang, S. L., and Tsai, L. W., 1993, ‘‘On the Redundant-Drive Backlash-Free Robotic Mechanisms,’’ ASME Transactions, JoMechanical Design, Vol. 115, No. 2, pp. 247–254.

ol. 125, SEPTEMBER 2003 Transactions of the ASME

://proceedings.asmedigitalcollection.asme.org/ on 07/27/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use

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39 Lin, C. C., and Tsai, L. W., 1993, ‘‘The Trajectory Analysis of Bevel Planetary Gear Trains,’’ ASME Journal of Mechanical Design, VoNo. 1, pp. 164–170.

40 Lee, J. J., and Tsai, L. W., 1991, ‘‘On the Structural Synthesis of Tendon-Driven Manipulators Having Pseudo-Triangular StructureInternational Journal of Robotics Research, Vol. 10, No. 3, pp. 255–262.

41 Tsai, L. W., and Lu, J. J., 1990, ‘‘Coupler-Point-Curve Synthesis using Homotopy Methods,’’ASME Journal of Mechanical Design, VNo. 3, pp. 384–389.

42 Chang, S. L., and Tsai, L. W., 1990, ‘‘Topological Synthesis of Articulated Gear Mechanisms,’’ IEEE Transactions on RobotiAutomation, Vol. 6, No. 1, pp. 97–103.

43 Azarm, S., Chen, R., and Tsai, L. W., 1990, ‘‘Walking Robot: A Multidisciplinary Design Project for Undergraduate Students,’’ InternJournal of Mechanical Engineering Education, Vol. 18, No. 2, pp. 101–115.

44 Tsai, L. W., and Lin, C. C., 1989, ‘‘The Creation of Non-fractionated, Two Degree-of-Freedom Epicyclic Gear Trains,’’ ASME JourMechanisms, Transmissions, and Automation in Design, Vol. 111, No. 4, pp. 524–529.

45 Tsai, L. W., and Maki, E. R., 1989, ‘‘Planetary-Gear-Type Second-Harmonic Balancers,’’ ASME Journal of Mechanisms, TransmissiAutomation in Design, Vol. 111, No. 4, pp. 530–536.

46 Freudenstein, F., Tsai, L. W., and Maki, E. R., 1989, ‘‘The Synthesis and Analysis of Variable-Valve-Timing Mechanisms for InCombustion Engines,’’ SAE Transactions, Vol. 97, pp. 6.642–6.651.

47 Tsai, L. W., and Freudenstein, F., 1989, ‘‘On the Conceptual Design of a Novel Class of Robot Configurations,’’ ASME JouMechanisms, Transmissions, and Automation in Design, Vol. 111, No. 1, pp. 47–53.

48 Tsai, L. W., and Lee, J. J., 1989, ‘‘Kinematic Analysis of Tendon-Driven Robotic Mechanisms using Graph Theory,’’ ASME JouMechanisms, Transmissions, and Automation in Design, Vol. 111, No. 1, pp. 59–65.

49 Tsai, L. W., Maki, E. R., and Jacques, R. C., 1988, ‘‘Evaluation of the Oldham-Coupling-Type Balancer on a 90-degree V-6 EngineTransactions, Vol. 96, pp. 4.10–4.15.

50 Tsai, L. W., 1988, ‘‘The Kinematics of Spatial Robotic Bevel-Gear Trains,’’ IEEE Transactions on Robotics and Automation, Vol. 4,pp. 150–156.

51 Tsai, L. W., 1987, ‘‘Discussion—On the Maximum Value of the Maximum Degree of Kinematic Chain,’’ ASME Journal of MechanTransmissions, and Automation in Design, Vol. 109, No. 4, p. 490.

52 Tsai, L. W., 1987, ‘‘An Application of Linkage Characteristic Polynomial to the Topological Synthesis of Epicyclic Gear Trains,’’ AJournal of Mechanisms, Transmissions, and Automation in Design, Vol. 109, No. 3, pp. 329–336.

53 Tsai, L. W., and Morgan, A. P., 1985, ‘‘Solving the Kinematics of the Most General Six- and Five-Degree-of-Freedom ManipulaContinuation Methods,’’ ASME Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 107, No. 2, pp. 189–200.

54 Tsai, L. W., and Walter, R., 1985, ‘‘Evaluation of the Oldham-Coupling Type Balancer on a 2.5 Liter In-line Four-Cylinder Engine,Transactions, Vol. 93, Paper No. 840456, pp. 3.378–3.382.

55 Freudenstein, F., Tsai, L. W., and Maki, E. R., 1984, ‘‘The Generalized Oldham Coupling,’’ ASME Journal of Mechanisms, Transmand Automation in Design, Vol. 106, No. 4, pp. 475–481.

56 Tsai, L. W., 1984, ‘‘Oldham Coupling Second-Harmonic Balancer,’’ ASME Journal of Mechanisms, Transmissions, and AutomaDesign, Vol. 106, No. 3, pp. 285–290.

57 Tsai, L. W., 1983, ‘‘Design of Drag-Link Mechanisms with Minimax Transmission Angle Deviation,’’ ASME Journal of MechaniTransmissions, and Automation in Design, Vol. 105, No. 4, pp. 686–691.

58 Tsai, L. W., 1983, ‘‘Design of Drag-Link Mechanisms with Optimum Transmission Angle,’’ ASME Journal of Mechanisms, Transmisand Automation in Design, Vol. 105, No. 2, pp. 254–258.

59 Tsai, L. W., 1982, ‘‘Kinematic Synthesis of an Adjustable Six-Bar Angular Motion Compensator,’’ASME Journal of Mechanical Desig104, No. 1, pp. 21–28.

60 Tsai, L. W., and Ciardella, R. L., 1979, ‘‘Linear Step Motor Design Provides High Plotter Performance at Low Cost,’’ Hewlett-PJournal, pp. 7–14.

61 Tsai, L. W., 1977, ‘‘Instantaneous Kinematics of a Special Two-Parameter Motion,’’ ASME Journal of Engineering for Industry, VSeries B, No. 1, pp. 336–340.

62 Tsai, L. W., and Roth, B., 1973, ‘‘A Note on the Design of Revolute-Revolute Cranks,’’ Journal of Mechanism and Machine Theory,pp. 23–31.

63 Tsai, L. W., and Roth, B., 1973, ‘‘Incompletely Specified Displacement: Geometry and Spatial Linkage Synthesis,’’ ASME JouEngineering for Industry, Vol. 95, Series B, No. 2, pp. 603–611.

64 Tsai, L. W., and Roth, B., 1972, ‘‘Design of Dyads with Helical, Cylindrical, Spherical, Revolute, and Prismatic Joints,’’ JourMechanism and Machine Theory, Vol. 7, No. 1, pp. 85–102.

65 Reismann, H., and Tsai, L. W., 1972, ‘‘Forced Motion in Elastic Cylindrical Rods—A Comparison of Two Theories,’’ ZeitschriAngewandte Mathematik & Mechanik, Vol. 52, pp. 565–572.

66 Reismann, H., and Tsai, L. W., 1972, ‘‘Wave Propagation in Discretely Inhomogeneous Elastic Cylindrical Rods—A ComparisonTheories,’’ Zeitschrift fur Angewandte Mathematik & Mechanik, Vol. 52, pp. 1–10.

of Mechanical Design SEPTEMBER 2003, Vol. 125 Õ 649

://proceedings.asmedigitalcollection.asme.org/ on 07/27/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use