Abstract
The purpose of this paper is to study the nonstationary vibration of a string with time-varying length and a mass-spring system attached at the lower end. The string is hung vertically and excited sinusoidally by a horizontal displacement at its upper end. The mass is supported by a guide spring horizontally and has two-degrees-of-freedom, vertical and horizontal. It is shown analytically that axial velocity of the string influences the peak amplitude of the string vibration at the passage through resonances. Moreover, it is shown numerically that the amplitudes of both the string and the mass vibrations depend on the sign of the axial velocity, when the natural frequency of the mass-spring system is close to the frequency of the excitation. The above two theoretical results are confirmed experimentally with a simple experimental setup.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Carrier, G. F., ‘The spaghetti problem’, The American Mathematical Monthly 56, 1949, 660–672.
Yamamoto, T., Yasuda, K., and Kato, M., ‘Vibration of a string with time-variable length’, Transactions of the Japanese Society of Mechanical Engineers 44–380c, 1978, 17–24 [in Japanese].
Mansfield, L. and Simmonds, J. G., ‘The reverse spaghetti problem: Drooping motion of an elastica issuing from a horizontal guide’, Journal of Applied Mechanics 54, 1987, 147–150.
Wickert, J. A. and Mote, C. D., Jr., ‘Classical vibration analysis of axially moving continua’, Journal of Applied Mechanics 57, 1990, 738–744.
Ram, Y. M. and Blech, J. J., ‘The dynamic behavior of a vibratory system after modification’, Journal of Sound and Vibration 150(3), 1991, 357–370.
Sasaki, Y., Teshima, N., and Ohnishi, S., ‘Vibration of elevator rope and cab in high-rise building’, Transactions of the Japanese Society of Mechanical Engineers 730–14, 1973, 165–168 [in Japanese].
Miles, J., ‘Resonant, nonplanar motion of a stretched string’, Journal of Acoustical Society of America 75(5), 1984, 1505–1510.
O'Reilly, O. and Holmes, P. J., ‘Non-linear, non-planar and non-periodic vibration of a string’, Journal of Sound and Vibration 153(3), 1992, 413–435.
Kevorkian, J. and Cole, J. D., Perturbation Methods in Applied Mathematics, Springer-Verlag, New York, 1981.
Nayfeh, A. H., Perturbation Methods, Wiley-Interscience Publishers, New York, 1973.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Terumichi, Y., Ohtsuka, M., Yoshizawa, M. et al. Nonstationary Vibrations of a String with Time-Varying Length and a Mass-Spring Attached at the Lower End. Nonlinear Dynamics 12, 39–55 (1997). https://doi.org/10.1023/A:1008224224462
Issue Date:
DOI: https://doi.org/10.1023/A:1008224224462