DOI QR코드

DOI QR Code

Natural Frequency of 2-Dimensional Heaving Circular Cylinder: Time-Domain Analysis

상하동요하는 2차원 원주의 고유진동수: 시간 영역 해석

  • Kim, Ki-Bum (Dept. Naval Architecture & Ocean Eng., Chungnam National University) ;
  • Lee, Seung-Joon (Dept. Naval Architecture & Ocean Eng., Chungnam National University)
  • 김기범 (충남대학교 선박해양공학과) ;
  • 이승준 (충남대학교 선박해양공학과)
  • Received : 2012.11.27
  • Accepted : 2013.06.28
  • Published : 2013.08.20

Abstract

The concept of the natural frequency is useful for understanding the characters of oscillating systems. However, when a circular cylinder floating horizontally on the water surface is heaving, due to the hydrodynamic forces, the system is not governed by the equation like that of the harmonic one. In this paper, in order to shed some lights on the more correct use of the concept of the natural frequency, a problem of the heaving circular cylinder is analyzed in the time domain. The equation of motion, an integro-differential equation, was derived following the fashion of Cummins (1962), and its coefficients including the retardation function were obtained using the numerical solution of Lee (2012). The equation was solved numerically, and the experiment was also carried out in the CNU flume. Using our numerical and experimental results, the natural frequency was defined as its average value given by the motion data excluding those of the initial stage. Our results were then compared with those of the existing investigations such as Maskell and Ursell (1970), Ito (1977) and Yeung (1982) as well as the newly obtained results of Lee (2012). Comparison showed that the natural frequency obtained here agrees well with that of Lee (2012), which was found through the frequency domain analysis. It was also shown that the approximation of heaving motion by a damped harmonic oscillation, which was regarded as suitable by most previous investigators, is not physically suitable for the reason that can be clearly shown through comparing the shape of MCFRs(Modulus of Complex Frequency Response). Furthermore, we found that although the previous approximations yield the damping ratio significantly different from our result the magnitude of natural frequency is not much different from our result.

Keywords

References

  1. Cummins, W.E., 1962. The Impulse Response Function and Ship Motions, DTMB Report 1661.
  2. Greenberg, M.D., 1978. Foundations of Applied Mathematics. Prentice-Hall Inc.
  3. Ito, S., 1977. Study of the transient heave oscillation of a floating cylinder. M.S. Thesis. Massachusetts Institute of Technology.
  4. Kim, K.B., 2012. Natural frequency of 2-dimensional heaving circular cylinder; time-domain analysis. M.S. Thesis. Chungnam National University.
  5. Lee, D.Y., 2012. Natural frequency of 2-dimensional heaving circular cylinder; frequency-domain analysis. M.S. Thesis. Chungnam National University.
  6. Maskell, S.J. & Ursell, F., 1970. The Transient Motion of a Floating Body. Journal of Fluid Mechanics, 44(2), pp.303-313. https://doi.org/10.1017/S0022112070001842
  7. Meirovitch, L., 1967. Analytical methods in vibrations. Macmillan Company: New York.
  8. Oortmerssen, G.V., 1976. The motion of a moored ship in waves. Ph.D. Dissertation. Delft University of Technology.
  9. Ursell, F., 1949. On the Heaving Motion of a Circular Cylinder on the Surface of a Fluid. Quarterly Journal of Mechanics & Applied Mathematics, 2(2), pp.218-231. https://doi.org/10.1093/qjmam/2.2.218
  10. Ursell, F., 1959. On the Virtual Mass and Damping of Floating Bodies at Zero Speed Ahead. Proceedings of Behavior of Ships in a Seaway, Wageningen, 7-10 September 1957, pp.374-387.
  11. Ursell, F., 1964. The decay of the free motion of a floating body. Journal of Fluid Mechanics, 19(2), pp.305-319. https://doi.org/10.1017/S0022112064000738
  12. Ursell, F., 2007. Water-wave Problems, Their Mathematical Solution and Physical Interpretation. Journal of Engineering Mathematics, 58(1-4), pp.7-17. https://doi.org/10.1007/s10665-006-9092-8
  13. Yeung, R.W., 1982. The Transient Heaving Motion of Floating Cylinders. Journal of Engineering Mathematics, 16(2), pp.97-119. https://doi.org/10.1007/BF00042549