DOI QR코드

DOI QR Code

Floquet 이론과 섭동법에 의한 Mathieu Equation의 안정성해석

Stability Analysis of Mathieu Equation by Floquet Theory and Perturbation Method

  • Park, Chan Il (Department of Precision Mehanical Engineering, Gangneung-Wonju National University)
  • 투고 : 2013.05.15
  • 심사 : 2013.07.18
  • 발행 : 2013.08.20

초록

In contrast of external excitations, parametric excitations can produce a large response when the excitation frequency is away from the linear natural frequencies. The Mathieu equation is the simplest differential equation with periodic coefficients, which lead to the parametric excitation. The Mathieu equation may have the unbounded solutions. This work conducted the stability analysis for the Mathieu equation, using Floquet theory and numerical method. Using Lindstedt's perturbation method, harmonic solutions of the Mathieu equation and transition curves separating stable from unstable motions were obtained. Using Floquet theory with numerical method, stable and unstable regions were calculated. The numerical method had the same transition curves as the perturbation method. Increased stable regions due to the inclusion of damping were calculated.

키워드

참고문헌

  1. Nayfeh, A. H. and Mook, D. T., 1979, Nonlinear Oscillations, John Wiley & Sons, Inc., New York.
  2. Meirovitch, L., 1970, Methods of Analytical Dynamics, McGraw-Hill, New York.
  3. Kim, W. S., Lee, D.-J. and Chung, J. T., 2005, Three-dimensional Modeling and Dynamic Analysis of an Automatic Ball Balancer in an Optical Disk Drive, Journal of Sound and Vibration, Vol. 285, No. 3, pp. 547-569. https://doi.org/10.1016/j.jsv.2004.08.016
  4. Mondo, M. and Cederbaum, G., 1993, Stability Analysis of the Non-linear Mathieu Equation, Journal of Sound and Vibration, Vol. 167, No. 1, pp. 77-89. https://doi.org/10.1006/jsvi.1993.1322
  5. Cho, Y. S. and Choi, Y. S., 2002, Nonlinear Dynamic Characteristics of Gear Driving Systems with Periodic Meshing Stiffness Variation and Backlash, Transactions of the Korean Society for Noise and Vibration Engineering, Vol. 12, No. 12, pp. 921-928. https://doi.org/10.5050/KSNVN.2002.12.12.921
  6. Sika, G. and Velex, P., 2008, Instability Analysis in Oscillators with Velocity-modulated Time-varying Stiffness-applications to Gears Submitted to Engine Speed Fluctuations, Journal of Sound and Vibration, Vol. 318, No. 1-2, pp. 166-175. https://doi.org/10.1016/j.jsv.2008.04.008
  7. Chen, S.-Y. and Tang, J.-Y., 2008, Study on a New Nonlinear Parametric Excitation Equation: Stability and Bifurcation, Journal of Sound and Vibration, Vol 318, No. 4-5, pp. 1109-1118. https://doi.org/10.1016/j.jsv.2008.04.055
  8. Stoker, J. J., 1950, Nonlinear Vibrations, Interscience Publishers, New York.
  9. Rao, S. S., 2004, Mechanical Vibrations, 4th ed. Pearson Education Inc., New Jersey.
  10. Park, C. I., 2013, Stability Analysis of Mathieu Equation by Floquet Theory, Proceedings of the KSNVE Annual Spring Conference, pp. 267-268.