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Article

Analysis and Control of Transverse Vibrations of Axially Moving Strings

[+] Author and Article Information
Li-Qun Chen

Department of Mechanics, Shanghai University, Shanghai 200444, China Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072, Chinae-mail: lqchen@online.sh.cn

Appl. Mech. Rev 58(2), 91-116 (Apr 06, 2005) (26 pages) doi:10.1115/1.1849169 History: Online April 06, 2005
Copyright © 2005 by ASME
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References

Skutch,  R., 1897, “Uber die Bewegung eines gespannten Fadens, welcher gezwungun ist, durch zwei feste Punkte mit einer constanten Geschwindigkeit zu gehen, und zwischen denselben in transversalen Schwingungen von gerlinger Amplitude versetzt wird,” Annalen der Physik und Chemie,61, pp. 190–195.
Aiken,  J., 1878, “An Account of Some Experiments on Rigidity Produced by Centrifugal Force,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science,5(29), pp. 81–105.
Mote,  C. D., 1972, “Dynamic Stability of Axially Moving Materials,” Shock Vib. Dig., 4(4), pp. 2–11.
Ulsoy,  A. G., Mote,  C. D., and Szymani,  R., 1978, “Principal Developments in Band Saw Vibration and Stability Research,” Holz Roh-Werkst., 36, pp. 273–280.
Wickert,  J. A., and Mote,  C. D., 1988a, “Current Research on the Vibration and Stability of Axially-Moving Materials,” Shock Vib. Dig., 20(5), pp. 3–13.
Wang,  K. W., and Liu,  S. P., 1991, “On the Noise and Vibration of Chain Drive Systems,” Shock Vib. Dig., 23(4), pp. 8–13.
Abrate,  A. S., 1992, “Vibration of Belts and Belt Drives,” Mech. Mach. Theory, 27(6), pp. 645–659.
Swope,  R. D., and Ames,  W. F., 1963, “Vibrations of a Moving Threadline,” J. Franklin Inst., 275(1), pp. 36–55.
Archibald,  F. R., and Emslie,  A. G., 1958, “The Vibration of a String Having a Uniform Motion Along Its Length,” J. Appl. Mech., 25(3), pp. 347–348.
Pakdemirli,  M., and Boyaci,  H., 2002, “Effect of Nonideal Boundary Conditions on the Vibrations of Continuous Systems,” J. Sound Vib., 249(4), pp. 815–823.
Sack,  R. A., 1954, “Transverse Oscillations in Travelling Strings,” Br. J. Appl. Phys., 5, pp. 224–226.
Mahalingam,  S., 1957, “Transverse Vibrations of Power Transmission Chains,” Br. J. Appl. Phys., 8, pp. 145–148.
Chubachi,  T., 1958, “Lateral Vibration of Axially Moving Wire or Belt Materials,” Bull. JSME, 1(1), pp. 24–29.
Miranker,  W. L., 1960, “The Wave Equation in a Medium in Motion,” IBM J. Res. Dev., 4(1), pp. 36–42.
Meirovitch, L., 1967, Analytical Methods in Vibrations, MacMillan, New York.
Meirovitch, L., 1997, Principles and Techniques of Vibrations, Prentice–Hall, New Jersey.
Meirovitch,  L., 1974, “A New Method of Solution of the Eigenvalue Problem for Gyroscopic Systems,” AIAA J., 12, pp. 1337–1342.
Meirovitch,  L., 1975, “A Model Analysis for the Response of Linear Gyroscopic Systems,” J. Appl. Mech., 42(2), pp. 446–450.
D’Eleuterio,  G. M., and Hughes,  P. C., 1984, “Dynamics of Gyroelastic Continua,” J. Appl. Mech., 51(2), pp. 415–422.
Hughes,  P. C., and D’Eleuterio,  G. M. T., 1986, “Modal Parameter Analysis of Gyroelastic Continua,” J. Appl. Mech., 53, pp. 918–924.
Wickert,  J. A., and Mote,  C. D., 1990, “Classical Vibration Analysis of Axially Moving Continua,” J. Appl. Mech., 57(3), pp. 738–744.
Wickert,  J. A., and Mote,  C. D., 1991, “Response and Discretization Method for Axially Moving Materials,” Appl. Mech. Rev., 44(11), pp. S279–284.
Jha,  R. K., and Parker,  R. G., 2000, “Spatial Discretization of Axially Moving Media Vibration Problems,” J. Vibr. Acoust., 122, pp. 290–294.
Renshaw,  A. A., and Mote,  C. D., 1996, “Local Stability of Gyroscopic Systems Near Vanishing Eigenvalues,” J. Appl. Mech., 63(1), pp. 116–120.
Kao, I., Wei, S., and Chiang, F. P., 1998, “Vibration Analysis of Wiresaw Manufacturing Processes and Wafer Surface Measurements,” NSF Des Manuf Grantees Conf, pp. 427–429.
Wei,  S., and Kao,  I., 2000, “Vibration Analysis of Wire and Frequency Response in the Modern Wiresaw Manufacturing Process,” J. Sound Vib., 231(5), pp. 1383–1395.
Tan,  C. A., and Chung,  C. H., 1993, “Transfer Function Formulation of Constrained Distributed Parameter Systems, Part 1: Theory,” J. Appl. Mech., 60(4), pp. 1004–1011.
Chung,  C. H., and Tan,  C. A., 1993, “Transfer Function Formulation of Constrained Distributed Parameter Systems, Part 2: Applications,” J. Appl. Mech., 60(4), pp. 1012–1019.
Perkins,  N. C., 1990, “Linear Dynamics of a Translating String on an Elastic Foundation,” J. Vibr. Acoust., 112(1), pp. 2–7.
Wickert,  J. A., 1994, “Response Solutions for the Vibration of a Traveling String on an Elastic Foundation,” J. Vibr. Acoust., 116(1), pp. 137–139.
Xiong,  Y., and Hutton,  S. G., 1994, “Vibration and Stability Analysis of a Multi-Guided Rotating String,” J. Sound Vib., 169(5), pp. 669–683.
Tan,  C. A., and Zhang,  L., 1994, “Dynamic Characteristics of a Constrained String Translating Across an Elastic Foundation,” J. Vibr. Acoust., 116, pp. 318–325.
Yang,  B., and Tan,  C. A., 1992, “Transfer Functions of One-Dimensional Distributed Parameter Systems,” J. Appl. Mech., 59(4), pp. 1009–1014.
Parker,  P. G., 1998, “On the Eigenvalues and Critical Speed Stability of Gyroscopic Continua,” J. Appl. Mech., 65, pp. 134–140.
Saeed,  H. M., and Festroni,  F., 1998, “Simulation of Combined System by Periodic Structures: The Wave Transfer Matrix Approach,” J. Sound Vib., 213(1), pp. 53–73.
Riedel,  C. H., and Tan,  C. A., 1998, “Dynamic Characteristics and Mode Localization of Elastically Constrained Axially Moving Strings and Beams,” J. Sound Vib., 215(3), pp. 455–473.
Parker,  P. G., 1999, “Supercritical Speed Stability of the Trivial Equilibrium of an Axially-Moving String on an Elastic Foundation,” J. Sound Vib., 221(2), pp. 205–219.
Cheng,  S. P., and Perkins,  N. C., 1991, “The Vibration and Stability of a Friction-Guided Translating String,” J. Sound Vib., 144(2), pp. 281–292.
Huang,  F. Y., and Mote,  C. D., 1995, “On the Translating Damping Caused by a Thin Viscous Fluid Layer Between a Translating String and Translating Rigid Surface,” J. Sound Vib., 181(2), pp. 251–260.
Chen,  J. S., 1997, “Natural Frequencies and Stability of an Axially Traveling String in Contact With a Stationary Load System,” J. Vibr. Acoust., 119(2), pp. 152–157.
Tan,  C. A., and Ying,  S., 1997, “Dynamic Analysis of the Axially Moving String Based on Wave Propagation,” J. Appl. Mech., 64, pp. 394–400.
Saeed,  H. M., 1999, “Disturbance Propagation Paths in a Constrained Axially Moving String,” J. Sound Vib., 228(1), pp. 218–225.
Le-Ngoc,  L., and McCallion,  H., 1999, “Dynamic Stiffness of an Axially Moving String,” J. Sound Vib., 220(4), pp. 749–756.
Rayleigh, J. W. S., 1945, The Theory of Sound, Dover, New York, Vol. 1.
Yang,  B., 1992, “Eigenvalue Inclusion Principles for Distributed Gyroscopic Systems,” J. Appl. Mech., 59(3), pp. 650–656.
Wickert,  J. A., and Mote,  C. D., 1988b, “Linear Transverse Vibration of an Axially Moving String-Particle System,” J. Acoust. Soc. Am., 84(3), pp. 963–969.
Wickert,  J. A., and Mote,  C. D., 1991a, “Traveling Load Response of an Axially Moving String,” J. Sound Vib., 149(2), pp. 267–284.
Lee,  K. Y., and Renshaw,  A. A., 2000, “Solution of Moving Mass Problem Using Complex Eigenfunction Expansions,” J. Appl. Mech., 67(6), pp. 823–827.
Renshaw,  A. A., 1997, “Modal Decoupling of Systems Described by Three Linear Operators,” J. Appl. Mech., 64(2), pp. 238–240.
Zhu,  W. D., and Mote,  C. D., 1994, “Free and Forced Response of an Axially Moving String Transporting a Damped Linear Oscillator,” J. Sound Vib., 177(5), pp. 591–610.
Zhu, W. D., and Mote, C. D., Jr., 1998, “On the Transient Response of Distributed Structures Interacting With Discrete Components,” in Dynamics and Control of Distributed Systems, edited by H. S. Tzou and L. A. Bergman, Cambridge University Press, Cambridge, pp. 1–68.
Tan,  C. A., Yang,  B., and Mote,  C. D., 1990, “On the Vibration of a Translating String Coupled to Hydrodynamic Bearings,” J. Vibr. Acoust., 112(3), pp. 337–345.
Mote,  C. D., 1965, “A Study of Band Saw Vibration,” J. Franklin Inst., 279(6), pp. 430–444.
Hwang,  S. J., Perkins,  N. C., Ulsoy,  A. G., and Mechstroth,  R. J., 1994, “Rotational Response and Slip Prediction of Serpentine Belt Drive Systems,” J. Vibr. Acoust., 116(1), pp. 71–78.
Kraver,  T. C., Fan,  G. W., and Shah,  J. J., 1996, “Complex Model Analysis of a Flat Belt Pulley System With Belt Damping and Coulomb-Damped Tensioner,” J. Mech. Des., 118(2), pp. 306–311.
Beikmann,  R. S., Perkins,  N. C., and Ulsoy,  A. G., 1996a, “Free Vibration of Serpentine Belt Drive Systems,” J. Vibr. Acoust., 118(3), pp. 406–413.
Zhang  L., and Zu  J. W., 1999, “Modal Analysis of Serpentine Belt Drive Systems,” J. Sound Vib., 222, pp. 259–279.
Zu, J. W., and Zhang, L., 2000, “Modal Analysis in Coupled Vibration of Serpentine Belt Drive System,” in Dyn Acoust Simulations, ASME DE Vol. 108 and DSC Vol. 68, pp. 177–183.
Zhang,  L., Zu,  J. W., and Hou,  Z., 2001, “Complex Modal Analysis of Non-Self-Adjoint Hybrid Serpentine Belt Drive Systems,” J. Vibr. Acoust., 123, pp. 150–156.
Zhu,  W. D., and Mote,  C. D., 1995, “Propagation of Boundary Disturbances in an Axially Moving Strip in Contact With Rigid and Flexible Constraints,” J. Appl. Mech., 62(4), pp. 873–879.
Zhu,  W. D., Mote,  C. D., and Guo,  B. Z., 1997, “Asymptotic Distribution of Eigenvalues of a Constrained Translating String,” J. Appl. Mech., 64(3), pp. 613–619.
Lakshmikumaran,  A. V., and Wickert,  J. A., 1996, “On the Vibrating of Coupled Traveling String and Air Bearing Systems,” J. Vibr. Acoust., 118(3), pp. 398–405.
Roos,  J. P., Schweigman,  C., and Timman,  R., 1973, “Mathematical Formulation of the Laws of Conservation of Mass and Energy and the Equation of Motion for a Moving Thread,” J. Eng. Math., 7(2), pp. 139–146.
Wickert,  J. A., and Mote,  C. D., 1989, “On the Energetics of Axially Moving Continua,” J. Acoust. Soc. Am., 85(3), pp. 1365–1368.
Renshaw,  A. A., 1997b, “The Energetics of Winched Strings,” J. Vibr. Acoust., 119(4), pp. 643–644.
Lee,  S. Y., and Mote,  C. D., 1997, “A Generalized Treatment of the Energetics of Translating Continua, Part 1: Strings and Second Order Tensioned Pipes,” J. Sound Vib., 204(5), pp. 717–734.
Lee,  S. Y., and Mote,  C. D., 1998, “Traveling Wave Dynamics in a Translating String Coupled to Stationary Constraints: Energy Transfer and Mode Localization,” J. Sound Vib., 212(1), pp. 1–22.
Renshaw,  A. A., Rahn,  C. D., Wickert,  J. A., and Mote,  C. D., 1998, “Energy and Conserved Functionals for Axially Moving Materials,” J. Vibr. Acoust., 120(2), pp. 634–636.
Mote,  C. D., 1966, “On the Nonlinear Oscillation of an Axially Moving String,” J. Appl. Mech., 33, pp. 463–464.
Thurman,  A. L., and Mote,  C. D., 1969, “Free, Periodic, Nonlinear Oscillation of an Axially Moving Strip,” J. Appl. Mech., 36(1), pp. 83–91.
Ames,  W. F., Lee ,  S. Y., and Zaiser,  J. N., 1968, “Nonlinear Vibration of a Traveling Threadline,” Int. J. Non-Linear Mech., 3, pp. 449–469.
Lee,  S. Y., 1969, “On the Equation of Motion of a Moving Threadline,” Dev. Mech.,5(3), pp. 543–561.
Kim,  Y. I., and Tabarak,  B., 1972, “On the Nonlinear Vibration of Travelling Strings,” J. Franklin Inst., 293, pp. 381–399.
Koivurova,  H., and Salonen,  E. M., 1999, “Comments on Nonlinear Formulations for Travelling String and Beam Problems,” J. Sound Vib., 225(5), pp. 845–856.
Courant, R., and Hilbert, D., 1953, Methods of Mathematical Physics, Wiley, New York, Vol. 2.
Ames,  W. F., and Vicario,  A. A., 1969, “On the Longitudinal Wave Propagation on a Traveling Threadline,” Dev. Mech.,5(3), pp. 733–746.
Ames,  W. F., Lee,  S. Y., and Vicario,  A. A., 1970, “Longitudinal Wave Propagation on a Traveling Threadline II,” Int. J. Non-Linear Mech., 5(3), pp. 413–426.
Lax,  P. F., 1964, “Development of Singularities of Solutions of Homogeneous Nonlinear Hyperbolic Equations,” J. Math. Phys., 5, pp. 611–613.
Jeferey,  A., 1967, “The Evolution of Discontinuities in Solutions of Homogeneous Non-Linear Hyperbolic Equations Having Smooth Initial Data,” J. Math. Mech., 17, pp. 331–333.
Bogoliuboy, N. N., and Mitropolsky, Y. A., 1961, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York.
Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillations, Wiley, New York.
Bapat,  V. A., and Srinivasan,  P., 1967, “Nonlinear Transverse Oscillations on Travelling Strings by the Method of Harmonic Balance,” J. Appl. Mech., 34, pp. 775–777.
Bapat,  V. A., and Srinivasan,  P., 1971, “Nonlinear Transverse Oscillations on Traveling Strings by a Direct Linearization Method,” J. Indian Indust. Sci.,53, pp. 120–125.
Mote,  C. D., and Thurman,  A. L., 1971, “Oscillation Modes of an Axially Moving Material,” J. Appl. Mech., 38, pp. 279–280.
Ghangrekar,  S. G., 1975, “On Nonlinear Oscillation of Moving String,” Labdev J. Sci. Tech: Phys. Sci. A,13, pp. 255–257.
Korde,  K. R., 1985, “On Nonlinear Oscillation of Moving String,” J. Appl. Mech., 52, pp. 493–494.
Burton,  T. D., and Hamdan,  M. N., 1983, “Analysis of Nonlinear Autonomous Conservative Oscillators by a Time Transformation Method,” J. Sound Vib., 87(4), pp. 543–554.
Nayfeh,  A. H., Nayfeh,  J. F., and Mook,  D. T., 1992, “On Methods for Continuous Systems With Quadratic and Cubic Nonlinearities,” Nonlinear Dyn., 3(1), pp. 145–162.
Pakdemirli,  M., 1994, “A Comparison of Two Perturbation Methods for Vibrations of Systems With Quadratic and Cubic Nonlinearities,” Mech. Res. Commun., 21, pp. 203–208.
Pakdemirli,  M., and Boyaci,  H., 1995, “Comparison of Direct-Perturbation Methods With Discretization-Perturbation Methods for Nonlinear Vibrations,” J. Sound Vib., 186(5), pp. 837–845.
Pakdemirli,  M., Nayfeh,  S. A., and Nayfeh,  A. H., 1995, “Analysis of One-to-One Autoparametric Resonances in Cables: Discretization vs Direct Treatment,” Nonlinear Dyn., 8(1), pp. 65–83.
Nayfeh, A. H., Nayfeh, S. A., and Pakdemirli, M., 1995, “On the Discretization of Weakly Nonlinear Spatially Continuous Systems,” in Nonlinear Dynamic Stochastic Mechanism, edited by N. S. Namachchivaya and W. Kliemann, CRC Press, Boca Raton.
Wickert,  J. A., 1993, “Analysis of Self-Excited Longitudinal Vibration of a Moving Tape,” J. Sound Vib., 160(3), pp. 455–463.
Majewski,  T., 1986, “Audio Signal Modulation Caused by Self-Excited Vibrations of Magnetic Tape,” J. Sound Vib., 105(1), pp. 17–25.
Jones, D. I., 2001, Handbook of Viscoelastic Vibration Damping, Wiley, New York.
Zhang,  L., and Zu,  J. W., 1998, “Nonlinear Vibrations of Viscoelastic Moving Belts, Part 1: Free Vibration Analysis,” J. Sound Vib., 216(1), pp. 75–91.
Hou,  Z., and Zu,  J. W., 2002, “Nonlinear Free Oscillations of Moving Belts With Standard Viscoelastic Model,” Mech. Mach. Theory, 37(9), pp. 925–940.
Christensen, R. M., 1982, Theory of Viscoelasticity, Academic, New York.
Hou, Z., and Zu, J. W., 2001, “Nonlinear Free Oscillations of Viscoelastic Moving Belts Using Maxwell-Kelvin Model,” in Proceedings of the Asia-Pacific Vib Conference, Jilin Sci Tech Press, Changchun, pp. 39–43.
Zu, J. W., and Hou, Z., 2000, “Comparison of Different Viscoelastic Models for Nonlinear Free Vibrations of Moving Belts,” in Vib. Cont. Continuous Syst. (ASME DE), Vol. 107, pp. 55–61.
Moustafa,  M. A., and Salman,  F. K., 1976, “Dynamic Properties of a Moving Thread Line,” J. Eng. Ind., 98, pp. 868–875.
Chen, L. Q., Zhao, W. J., and Zu, J. W., 2005, “Simulations of Transverse Vibrations of an Axially Moving String: A Modified Difference Approach” Appl. Math. Comput. (accepted).
Brenan, K. E., Campbell, S. L., and Petzold, L. R., 1996, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, SIAM, Philadelphia.
Fung,  R. F., Wang,  Y. C., and Wu  J. W., 1999, “Group Properties and Group-Invariant Solutions for Infinitesimal Transformations of the Nonlinearly Traveling String,” Int. J. Non-Linear Mech., 34, pp. 693–698.
Chen,  L. Q., and Zu,  J. W., 2004, “Energetics and Conserved Functional of Moving Materials Undergoing Transverse Nonlinear Vibration,” J. Vib. Acoust. 126(3), pp. 452–455.
Chen,  L. Q., and Zhao,  W. J., 2005, “The Energetics and Stability of Axially Moving Kirchhoff Strings,” J. Acoust. Soc. Am., 117(5), pp. 55–58.
Shih,  L. Y., 1971, “Three Dimensional Nonlinear Vibration of a Travelling String,” Int. J. Non-Linear Mech., 6, pp. 427–434.
Shih,  L. Y., 1975, “Motion of Elliptic Ballooning for a Travelling String,” Int. J. Non-Linear Mech., 10(3/4), pp. 183–191.
Moon,  J., and Wickert,  J. A., 1997, “Nonlinear Vibration of Power Transmission Belts,” J. Sound Vib., 200(4), pp. 419–431.
Kirchhoff, G., 1877, Vorlesungen ueber Mathematische Physik: Mechanik, Druck und Verlag von B. G. Teubner, Leipzig.
Zhang,  L., and Zu,  J. W., 1999b, “Nonlinear Vibration of Parametrically Excited Moving Belts, Part 2: Stability Analysis,” J. Appl. Mech., 66(2), pp. 403–409.
Chen,  L. Q., and Zhao,  W. J., 2005, “A Numerical Method for Simulating Transverse Vibrations of Axially Moving Strings”, Appl. Math. Comput, 160(2), pp. 411–422.
Ariartnam,  S. T., and Asokanthan,  S. F., 1987, “Dynamic Stability of Chain Drives,” ASME J. Mech., Transm., Autom. Des., 109(3), pp. 412–418.
Mochensturm,  E. M., Perkins,  N. C., and Ulsoy,  A. G., 1996, “Stability and Limit Cycles of Parametrically Excited, Axially Moving Strings,” J. Vibr. Acoust., 116(3), pp. 346–351.
Mote,  C. D., 1968, “Parametric Excitation of an Axially Moving String,” J. Appl. Mech., 35(1), pp. 171–172.
Naguleswaran,  S., and Williams,  C. J. H., 1968, “Lateral Vibration of Band-Saw, Pulley Belts and the Like,” Int. J. Mech. Sci., 10, pp. 239–250.
Liu,  Z. S., and Huang,  C., 2002, “Evaluation of the Parametric Instability of an Axially Translating Media Using a Variational Principle,” J. Sound Vib., 257(5), pp. 985–999.
Rhodes,  J. E., 1970, “Parametric Self-Excitation of a Belt Into Transverse Vibration,” J. Appl. Mech., 37(4), pp. 1055–1060.
Mote,  C. D., 1975, “Stability of Systems Transporting Accelerating Axially Moving Materials,” J. Dyn. Syst., Meas., Control, 97, pp. 96–98.
Pakdemirli,  M., and Batan,  H., 1993, “Dynamic Stability of a Constantly Accelerating Strip,” J. Sound Vib., 168(2), pp. 371–378.
Pakdemirli,  M., Ulsoy,  A. G., and Ceranoglu,  A., 1994, “Transverse Vibration of an Axially Accelerating String,” J. Sound Vib., 169(2), pp. 179–196.
Pakdemirli,  M., and Ulsoy,  A. G., 1997, “Stability Analysis of an Axially Accelerating String,” J. Sound Vib., 203(5), pp. 815–832.
Öz,  H. R., Pakdemirli,  M., and Özkaya,  E., 1998, “Transition Behavior From String to Beam for an Axially Accelerating Material,” J. Sound Vib., 215(3), pp. 571–576.
Özkaya,  E., and Pakdenirli,  M., 2000, “Lie Group Theory and Analytical Solutions for the Axially Accelerating String Problem,” J. Sound Vib., 230(4), pp. 729–742.
Suweken,  G., and Van Horssen,  W. T., 2003, “On the Transversal Vibrations of a Conveyor Belt With a Low and Time-Varying Velocity. Part 1: The String-Like Case,” J. Sound Vib., 264, pp. 117–133.
Wickert,  J. A., 1996, “Transient Vibration of Gyroscopic Systems With Unsteady Superposed Motion,” J. Sound Vib., 195(5), pp. 797–807.
Hsu,  C. S., 1963, “On the Parametric Excitation of a Dynamic System Having Multiple Degrees of Freedom,” J. Appl. Mech., 30, pp. 367–372.
Huang,  J. S., Fung,  R. F., and Lin,  C. H., 1995, “Dynamic Stability of a Moving String Undergoing Three-Dimensional Vibration,” Int. J. Mech. Sci., 37(2), pp. 145–160.
Fung,  R. F., and Wu,  S. L., 1997, “Dynamic Stability of a Three-Dimensional String Subjected to Both Magnetic and Tensioned Excitations,” J. Sound Vib., 204(1), pp. 171–179.
Fung,  R. F., Huang,  J. S., and Yeh,  J. Y., 1998, “Nonlinear Dynamic Stability of a Moving String by Hamiltonian Formulation,” Comput. Struct., 66(5), pp. 597–612.
Pellican, F., Vestroni, F., and Fregolent, A., 2000, “Experimental and Theoretical Analysis of a Power Transmission Belt,” in Vib. Cont. Continuous Syst. (ASME DE), Vol. 107, pp. 71–78.
Nayfeh, A. H., 1993, Method of Normal Form, Wiley, New York.
Zhang,  L., and Zu,  J. W., 1999, “Nonlinear Vibration of Parametrically Excited Moving Belts, Part 1: Dynamic Response,” J. Appl. Mech., 66(2), pp. 396–402.
Zhang,  L., and Zu,  J. W., 1999, “Nonlinear Vibration of Parametrically Excited Moving Belts, Part 2: Stability Analysis,” J. Appl. Mech., 66(2), pp. 403–409.
Hou, Z., and Zu, J. W., 2001, “Parametric Vibration of Viscoelastic Moving Belts Using Standard Linear Solid Model,” ASME 18th Bieannial Conf. Mech. Vib. Noise, September 9–12, 2001, Pittsburgh.
Chen,  L. Q., and Zu,  J. W., 2003, “Parametric Resonance of Axially Moving String with an Integral Constitutive Law,” Int. J. Nonlinear Sci. Numer. Simul.,4(2), pp. 169–177.
Chen,  L. Q., Zu,  J. W., and Wu,  J., 2003, “Dynamic Response of the Parametrically Excited Axially Moving String Constituted by the Boltzmann Superposition Principle,” Acta Mech., 162, pp. 143–155.
Fung,  R. F., Huang,  J. S., and Chen,  Y. C., 1997, “The Transient Amplitude of the Viscoelastic Travelling String: An Integral Constitutive Law,” J. Sound Vib., 201(2), pp. 153–167.
Wu,  J., and Chen,  L. Q., 2004, “Steady-State Responses and Their Stability of Nonlinear Vibration of an Axially Accelerating String,” Appl. Math. Mech., 25(9), pp. 1001–1011.
Chen,  L. Q., Zu,  J. W., and Wu,  J., 2004, “Principal Resonance in Transverse Nonlinear Parametric Vibration of an Axially Accelerating Viscoelastic String,” Acta Mech. Sinica ), 20(3), pp. 307–316.
Chen,  L. Q., Zu,  J. W., Wu,  J., and Yang,  X. D., 2004, “Transverse Vibrations of an Axially Accelerating Viscoelastic String With Geometric Nonlinearity,” J. Eng. Math., 48(2), pp. 171–182.
Chen, L. Q., 2005, “Principal Parametric Resonance of Axially Accelerating Viscoelastic Strings Constituted by the Boltzmann Superposition Principle,” Proc. Royal. Soc. London, (revised).
Fung,  R. F., Huang  J. S., Chen,  Y. C., and Yao,  C,. M., 1998, “Nonlinear Dynamic Analysis of the Viscoelastic String With a Harmonically Varying Transport Speed,” Comput. Struct., 66(6), pp. 777–784.
Zhang,  L., Zu,  J. W., and Zhong,  Z., 2002, “Transient Response for Viscoelastic Moving Belts Using Block-by-Block Method,” International Journal of Structural and Dynamics,2(2), pp. 265–280.
Lize, P., 1985, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia.
Chen,  L. Q., Zhao,  W. J., and Zu,  J. W., 2004, “Transient Responses of an Axially Accelerating Viscoelastic String Constituted by a Fractional Differentiation Law” J. Sound Vib., 278(4.5), pp. 861–871.
Zhao, W. J., and Chen, L. Q., 2003, “Iterative Techniques for Simulating Transverse Vibration of Axially Moving Viscoelastic Strings With Integral Constitutive” (preprint).
Chung,  J., Han,  C. S., and Yi,  K., 2001, “Vibration of an Axially Moving String With Geometric Non-Linearity and Translating Acceleration,” J. Sound Vib., 240(4), pp. 733–746.
Chung,  J., and Hulbert,  G. M., 1993, “A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method,” J. Appl. Mech., 60, pp. 371–375.
Bhat,  R. B., Xistris,  G. D., and Sankar,  T. S., 1982, “Dynamic Behavior of a Moving Belt Supported on an Elastic Foundation,” J. Mech. Des., 104(1), pp. 143–147.
Zhao,  W. J., and Chen,  L. Q., 2002, “A Numerical Algorithm for Non-Linear Parametric Vibration Analysis of a Viscoelastic Moving Belt,” International Journal of Nonlinear Science and Numerical Simulation,3(2), pp. 139–144.
Chen,  L. Q., and Zhao,  W. J., 2005, “A Computation Method for Nonlinear Vibration of Axially Accelerating Visoelastic Strings,” Appl. Math. Comput., 162(1), pp. 305–310.
Argyris, J., Faust, G., and Haase, M., 1994, An Exploration of Chaos, North-Holland, Amsterdam.
Nayfeh, A. H., and Balachandran, B., 1995, Applied Nonlinear Dynamics: Analytical, Computational, and Experiment Methods, Wiley, New York.
Chen,  L. Q., Zhang,  N. H., and Zu,  J. W., 2002, “Bifurcation and Chaos of an Axially Moving Viscoelastic String,” Mech. Res. Commun., 29(2/3), pp. 81–90.
Chen,  L. Q., Zhang,  N. H., and Zu,  J. W., 2003, “The Regular and Chaotic Vibrations of an Axially Moving Viscoelastic String Based on 4-Order Galerkin Truncation,” J. Sound Vib., 261(4), pp. 764–773.
Chen,  L. Q., Wu,  J., and Zu,  J. W., 2004, “Asymptotic Nonlinear Behaviors in Transverse Vibration of an Axially Accelerating Viscoelastic String” Nonlinear Dyn., 35(4), pp. 659–666.
Chen,  L. Q., Wu,  J., and Zu,  J. W., 2003b, “The Chaotic Response of the Viscoelastic Traveling String: an Integral Constitutive Law” Chaos, Solitons, Fractals, 21(2), pp. 349–357.
Chen, L. Q., and Zhang, N. H., 2004, “Nonlinear Dynamics of Axially Moving Viscoelastic Strings Based on Translating Eigenfunctions,” XXI INternational Congress of Theoretical and Applied Mechanics, 15-21 August, 2001, Warsaw, Poland.
Leamy,  M. J., and Perkins,  N. C., 1998, “Nonlinear Periodic Response of Engine Accessory Drives With Dry Friction Tensioners,” J. Vibr. Acoust., 118(4), pp. 909–916.
Chen,  L. Q., and Wu,  Z. M., 2003, “Amplitude-Frequency Characteristic of Nonlinear Vibration of a Serpentine Belt Drive System,” Eng. Mech.,20(1), pp. 149–152 (in Chinese).
Chen,  L. Q., and Wu,  Z. M., 2002, “Averaging Method for Analyzing a Multi-Degree-of-Freedom Nonlinear Oscillation,” J. Vib. Shock,21(3), pp. 63–64 (in Chinese).
Zu, J. W., Jia, H. S., and Zhong, Z., 2002, “Dynamic Analysis of Automotive Serpentine Belt Drive Systems With Coulomb Friction” (preprint).
Cheng,  G., and Zu,  J. W., 2004, “Non-Stick and Stick-Slip Motion of a Coulomb-Damped Belt Drive Systems Subjected to Multi-Frequency Excitations,” J. Appl. Mech., 70(6), pp. 871–884.
Ulsoy,  A. G., Whitesell,  J. E., and Hooven,  M. D., 1985, “Design of Belt-Tensioner Systems for Dynamic Stability,” J. Vibr. Acoust., 107, pp. 282–290.
Wang,  K. W., 1992, “On the Stability of Chain Drive Systems Under Periodic Sprocket Oscillations,” J. Vibr. Acoust., 114(1), pp. 119–126.
Beikmann,  R. S., Perkins,  N. C., and Ulsoy,  A. G., 1996b, “Nonlinear Coupled Response of Serpentine Belt Drive Systems,” J. Vibr. Acoust., 118(4), pp. 567–574.
Zhang,  L., and Zu,  J. W., 2000, “One-to-One Auto-Parametric Resonance in Serpentine Belt Drive Systems,” J. Sound Vib., 232(4), pp. 783–806.
Zu,  J. W., and Zhang,  L., 2000, “Two-to-One Internal Resonance in Serpentine Belt Drive Systems,” Int. J. Nonlinear Sci. Numer. Simulat.,1(3), pp. 187–198.
Fung,  R. F., and Shieh,  J. S., 1997, “Vibration Analysis of a Non-Linear Coupled Textile-Rotor System With Synchronous Whirling,” J. Sound Vib., 199(2), pp. 207–221.
Fung,  R. F., and Cheng,  W. H., 1993, “Free Vibration of a String/Slider Nonlinear Coupling System,” J. Chinese Soc. Mech. Eng.,12, pp. 229–239.
Fung,  R. F., Lin,  J. H., and Yao,  C. M., 1997, “Vibration Analysis and Suppression Control of an Elevator String Actuated by a PM Synchronous Servo Motor,” J. Sound Vib., 206(3), pp. 399–423.
Fung,  R. F., Huang,  J. S., and Chu,  J. J., 1998, “Dynamic Stability of an Axially Traveling String/Slider Coupling System With Moving Boundary,” J. Sound Vib., 211(4), pp. 689–701.
Yao,  C. M., Fung,  R. F., and Tseng,  C. R., 1999, “Nonlinear Vibration Analysis of a Traveling String With Time-Dependent Length by New Hybrid Laplace Transform/Finite Element Method,” J. Sound Vib., 219(2), pp. 323–337.
Leung,  A. Y. T., 2000, “Comment on ‘Non-Linear Vibration Analysis of a Traveling String With Time-Dependent Length by New Hybrid Laplace Transform/Finite Element Method,’” J. Sound Vib., 235(5), pp. 877–878.
Fung,  R. F., and Chang,  H. C., 2001, “Dynamic and Energetic Analyses of a String-Slider Non-Linear Coupling System by Variable Grid Finite Difference,” J. Sound Vib., 239(3), pp. 505–514.
Zhu,  W. D., and Ni,  J., 2000, “Energetics and Stability of Translating Media With an Arbitrarily Varying Length,” J. Vibr. Acoust., 122, pp. 295–304.
Meirovitch, L., 1990, Dynamics and Control of Structures, Wiley, New York.
Banks, S. P., 1983, State-Space and Frequency-Domain Methods in the Control of Distributed Parameter System, Peter Peregrinus, London.
Hughes,  P. C., and Skelton,  R. E., 1979, “Stability, Controllability and Observability of Linear Matrix-Second-Order Systems,” J. Appl. Mech., 47(2), pp. 415–420.
Juang,  J., and Balas,  M., 1978, “Dynamics and Control of Large Spinning Spacecraft,” J. Astronaut. Sci., 28(1), pp. 31–48.
Yang,  B., and Mote,  C. D., 1991, “Controllability and Observability of Distributed Gyroscopic Systems,” J. Dyn. Syst., Meas., Control, 113(1), pp. 11–17.
Jai,  A., and Pritchard,  A., 1987, “Sensors and Actuators in Distributed Systems,” Int. J. Control, 46(4), pp. 1139–1153.
Curtain, R. F., and Prichard, A. J., 1978, Infinite Dimensional Linear System Theory, Springer-Verlag, New York.
Sunar,  M., and Rao,  S. S., 1999, “Recent Advances in Sensing and Control of Flexible Structures via Piezoelectric Materials Technology,” Appl. Mech. Rev., 52(1), pp. 1–16.
Kostyuk,  V. I., Krasnoproshina,  A. A., and Ilyukhin,  A. G., 1983, “Vibration of a Longitudinally Moving String and Some Problem in Dynamics of Winding Sets,” Sov. Appl. Mech.,19(3), pp. 261–267.
Ulsoy,  A. G., 1984, “Vibration Control in Rotating or Translating Elastic Systems,” J. Dyn. Syst., Meas., Control, 106(1), pp. 6–14.
Butlovskiy, A. G., 1983, Structural Theory of Distributed Systems, Wiley, New York.
Yang,  B., 1992b, “Transfer Functions of Constrained/Combined One-Dimensional Continuous Dynamic Systems,” J. Sound Vib., 156(1), pp. 425–433.
Yang,  B., and Mote,  C. D., 1991, “Active Vibration Control of the Axially Moving String in the S Domain,” J. Appl. Mech., 58(1), pp. 189–196.
Yang,  B., and Mote ,  C. D., 1991, “Frequency-Domain Vibration Control of Distributed Gyroscopic System,” J. Dyn. Syst., Meas., Control, 113(1), pp. 18–25.
Yang,  B., and Mote,  C. D., 1992, “On Time Delay in Noncollocated Control of Flexible Mechanical Systems,” J. Dyn. Syst., Meas., Control, 114(3), pp. 409–415.
Yang,  B., 1992c, “Noncollocated Control of a Damped String Using Time Delay,” J. Dyn. Syst., Meas., Control, 114(4), pp. 736–740.
Vaughan,  D. R., 1968, “Application of Distributed Parameter Concepts to Dynamic Analysis and Control of Bending Vibrations,” J. Basic Eng., 90, pp. 157–166.
Flotow,  Ahvon, 1986, “Traveling Wave Control for Large Spacecraft Structures,” J. Guid. Control Dyn., 9(1), pp. 6–14.
MacMartin,  D. G., and Hall,  S. R., 1991, “Control of Uncertain Structures Using an H Power Flow Approach,” J. Guid. Control Dyn., 14(3), pp. 521–530.
Fujii,  H., Ohtsuka,  T., and Murayama,  T., 1992, “Wave-Absorbing Control for Flexible Structures With Noncollocated Sensors and Actuators,” J. Guid. Control Dyn., 15(2), pp. 431–439.
Chung,  C. H., and Tan,  C. A., 1995, “Active Vibration Control of the Axially Moving String by Wave Cancellation,” J. Vibr. Acoust., 117(1), pp. 49–55.
Ying,  S., and Tan,  C. A., 1996, “Active Vibration Control of the Axially Moving String Using Space Feedforward and Feedback Controllers,” J. Vibr. Acoust., 118(3), pp. 306–312.
Tan,  C. A., and Ying,  S., 2000, “Active Wave Control of the Axially Moving String: Theory and Experiment,” J. Sound Vib., 236(5), pp. 861–880.
Lee,  S. Y., and Mote,  C. D., 1996, “Vibration Control of an Axially Moving String by Boundary Control,” J. Dyn. Syst., Meas., Control, 118(1), pp. 66–74.
Fung,  R. F., Wu,  J. W., and Wu,  S. L., 1999a, “Exponential Stabilization of an Axially Moving String by Linear Boundary Feedback,” Automatica, 35, pp. 177–181.
Huang,  J. S., Wu,  J. W., and Lu,  P. Y., 2002, “A Study of Moving String With Partial State Feedback,” Int. J. Mech. Sci., 44(9), pp. 1893–1905.
Zhu,  W. D., Ni,  J., and Huang,  J., 2001, “Active Control of Translating Media With Arbitrarily Varying Length,” J. Vibr. Acoust., 123, pp. 347–358.
Zhang, F., Nagarkatti, S. P., Costic, B. T., Dawson, D. M., and Rahn, C. D., 1999, “Velocity Tracking Control of an Axially Accelerating String and Actuator System,” in Proceedings of the 38th Conf Decision Cont (IEEE), Vol. 5, pp. 4325–4330.
Nagarkatti  S. P., Zhang  F., Rahn  C. D., and Dawson  D. M., 2000, “Tension and Speed Regulation for Axially Moving Materials,” J. Dyn. Syst., Meas., Control, 122, pp. 445–453.
Slotine, J. J. E., and Li, W., 1991, Applied Nonlinear Control, Prentice–Hall, NJ.
Itkis, Y., 1976, Control System of Variable Structure, Wiley, New York.
Utkin, V. I., 1992, Sliding Modes in Control and Optimization, Springer-Verlag, New York.
Huang,  J. S., Fung,  F. R., and Chen,  D. S., 1994, “Application on Variable Structure Control in the Gyroscopic String Vibration,” Trans. Aeronaut Soc. Rep., China,26(4), pp. 329–338.
Fung,  R. F., and Liao,  C. C., 1995, “Application if Variable Structure Control in the Nonlinear String System,” Int. J. Mech. Sci., 37(9), pp. 985–993.
Meirovitch,  L., and Öz,  H., 1980, “Modal-Space Control of Distributed Gyroscopic Systems,” J. Guid. Control, 3(2), pp. 140–150.
Öz,  H., and Meirovitch,  L., 1980, “Optimal Modal-Space Control of Flexible Gyroscopic System,” J. Guid. Control Dyn., 3, pp. 218–226.
Meirocitch,  L., and Baruh,  H., 1983, “On the Problem of Observation Spillover in Self-Adjoint Distributed Parameter Systems,” J. Optim. Theory Appl., 39, pp. 269–275.
Fung,  R. F., Huang,  J. S., Wang,  Y. C., and Yang,  R. T., 1998, “Vibration Reduction of the Nonlinearly Traveling String by a Modified Variable Structure Control With Proportional and Integral Compensations,” Int. J. Mech. Sci., 40(6), pp. 493–506.
Chern,  T. L., and Wu,  Y. C., 1991, “Design of Integral Variable Structure Controller and Application to Electrohydraulic Velocity Servosystems,” IIE Trans., 138(5), pp. 439–444.
Orlov  Yu. V., and Utkin  V. I., 1987, “Sliding Mode Control in Indefinite-Dimensional Systems,” Automatica, 23(6), pp. 753–757.
Fung  R. F., and Tseng  C. C., 1999, “Boundary Control of an Axially Moving String via Lyapunov Method,” J. Dyn. Syst., Meas., Control, 121(1), pp. 105–110.
Queiroz, M. A., Dawson, D. M., Nagarkatti, S. P., and Zhang, F., 2000, Lyapunov-Based Control of Mechanical Systems, Birkhauser, Boston.
Shahruz,  S. M., and Kurmaji,  D. A., 1997, “Vibration Suppression of a Non-Linear Axially Moving String by Boundary Control,” J. Sound Vib., 201(1), pp. 145–152.
Shahruz,  S. M., 2000, “Boundary Control of a Nonlinear Axially Moving String,” Int. J. Robust Nonlinear Control, 10(1), pp. 7–25.
Komornik, V., 1994, Exact Controllability and Stabilization: The Energy Multiplier Method, Wiley, New York.
Luo, A. H., Guo, B. Z., and Morgul, O., 1999, Stability and Stabilization of Infinite Dimensional Systems With Applications, Stringer-Verlag, New York.
Shahruz,  S. M., 1998, “Boundary Control of the Axially Moving Kirchhoff String,” Automatica, 34(10), pp. 1273–1277.
Shahruz,  S. M., and Parasurama,  S. A., 1998, “Suppression of Vibration in the Axially Moving Kirchhoff String by Boundary Control,” J. Sound Vib., 214(3), pp. 567–575.
Fung,  R. F., Wu,  J. W., and Wu,  S. L., 1999b, “Stabilization of an Axially Moving String by Nonlinear Boundary Feedback,” J. Dyn. Syst., Meas., Control, 121(1), pp. 117–121.
Åström, K. J., and Wittenmark, B., 1995, Adaptive Control, Addison-Wesley, New York.
Krstic, M., Kanellakopoulos, I. K., and Kokotovic, P., 1995, Nonlinear and Adaptive Control Design, CRC Press, FL.
Queiroz,  M. S., Dawson,  D. M., Rahn,  C. D., and Zhang,  F., 1999, “Adaptive Vibration Control of an Axially Moving String,” J. Vibr. Acoust., 121(1), pp. 41–49.
Fung,  R. F., Wu,  J. W., and Lu,  P. Y., 2002, “Adaptive Boundary Control of an Axially Moving String System,” J. Vibr. Acoust., 124, pp. 435–440.
Middleton,  R. H., and Goodwin,  G. C., 1988, “Adaptive Computed Torque Control for Rigid Link Manipulator,” Syst. Control Lett., 10(1), pp. 9–16.
Spong,  M. W., and Ortega,  R., 1990, “On Adaptive Inverse Dynamics Control of Rigid Robots,” IEEE Trans. Autom. Control, 35(1), pp. 31–36.
Lammerts,  I. M. M., Veldpaus,  F. E., Molengraft Vande,  M. J. G., and Kok,  J. J., 1995, “Adaptive Reference Computed Torque Control of Flexible Robots,” J. Dyn. Syst., Meas., Control, 117(1), pp. 31–36.
Li,  Y., Aron,  D., and Rahn,  C. D., 2002, “Adaptive Vibration Isolation for Axially Moving Strings: Theory and Experiment,” Automatica, 38, pp. 379–390.
Huang,  J. S., Chao,  P. C. P., Fung,  R. F., and Lai,  C. L., 2003, “Parametric Control of an Axially Moving String via Fuzzy Sliding-Mode and Fuzzy Neural Network Methods,” J. Sound Vib., 264, pp. 177–201.
Rahn,  C. D., and Mote,  C. D., 1994, “Parametric Control of Flexible Systems,” J. Vibr. Acoust., 116, pp. 379–385.
Rahn,  C. D., and Mote,  C. D., 1996, “Parametric Control of Nonconservative Mechanical Systems,” J. Dyn. Syst., Meas., Control, 118, pp. 309–314.
Rahn,  C. D., and Mote,  C. D., 1996, “Axial Force Stabilization of Transverse Vibration in Pinned and Clamped Beams,” J. Dyn. Syst., Meas., Control, 118, pp. 379–380.
Chao,  P. C. P., and Lai,  C. L., 2003, “Boundary Control of an Axially Moving String via Fuzzy Sliding-Mode Control and Fuzzy Neural Network Methods,” J. Sound Vib., 262, pp. 795–813.
Rim,  W. T., and Kim,  K. J., 1994, “Identification of Tension in a Belt-Driven System by Analyzing Flexural Vibrations,” Mech. Syst. Signal Process., 8(2), pp. 199–213.
Qu,  Z., 2001, “Robust and Adaptive Boundary Control of a Stretched String on a Moving Transporter,” IEEE Trans. Autom. Control, 46(3), pp. 470–476.
Qu,  Z., 2002, “An Iterative Learning Algorithm for Boundary Control of a Stretched Moving String,” Automatica, 38, pp. 821–827.

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A prototypical three-pulley belt-driven system

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