Exact solution of linear parametrically excited system with transient frequency modulation
received 08.02.2002, published 29.04.2002
Download article (PDF, 400 kb, ZIP), use browser command "Save Target As..."
To read this document you need Adobe Acrobat © Reader software, which is simple to use and available at no cost. Use version 4.0 or higher. You can download software from Adobe site (http://www.adobe.com/).
To demonstrate the dangers of neglecting the transient characteristics of frequency modulation in linear parametrically excited systems a model equation with variable frequency that includes components of transient behavior is investigated. The transient components of the frequency modulation decay periodically and asymptotically with time leading to a periodic closed cycle form. The exact analytical solution of the model equation is derived for the first time in the present work by introducing a transformation that maps the original model into a system of two solvable equations. The extensive investigation of the general solution and its components demonstrates that the transient characteristics of the frequency variation cannot be ignored. In fact cases where theses characteristics can result in bounded or unbounded response of the system are presented. In general transient frequency characteristics continue to drastically affect the structure and the amplitude of the general solution long after they become indistinguishable in the frequency modulation. Numerical solutions of the model equation are also presented to further support the conclusions made in this paper.
15 pages, 12 figures
Сitation: Ayman A. Al-Maaitah. Exact solution of linear parametrically excited system with transient frequency modulation. Electronic Journal “Technical Acoustics”, http://www.ejta.org, 2002, 3.
1. Jaswal, J .S. and Bhave S. K., Experimental evaluation of damping in a bladed disk model. J. of Sound and Vibration, 177(4), 111-120, (1994).
2. Bently, D. E. and Muszynska, A. Perturbation study of a rotor/bearing system: identification of the oil whip resonance. ASME Design Engineering Division Conference and Exhibit on Mechanical Vibration and Noise, Cincinnati, Ohio, September 10-13, 189-198, (1985).
3. Kirk, R. G. and Gunter, E. J. Transient response of Rotor-Bearing Systems. ASME J. of Engineering for Industry, 81(2), 682-693, (1974).
4. Cveticanin, L. Self-Excited vibrations of the variable mass rotor/fluid system. J. of Sound and Vibration, 212(4), 685-702, (1998).
5. Alam, M. and Nelson, H. D. A blade loss response spectrum for flexible rotor systems. Transaction of the ASME J. of Engineering for Power, Paper No. 84-GT-29. (1983).
6. Nayfeh, A. H. and Mook, D. T. Nonlinear Oscillation. John Wiley and Sons, New York, (1979).
7. Mitropoloskii, Yu. A., and Van Dao, N. Applied Asymptotic Method in Non-linear Oscillations. Kluwer, Dordrecht, (1997).
8. Sanliturk, K. Y. and Ewins, D. J. Modeling two-dimensional friction contact and its application using harmonic balance method. 193(4), 511-523, (1996).
9. Csaba, C. and Anderson, M. Optimization of friction damper weight, simulation and experiments. ASME Turbo Expo 97, Orlando, FL, USA, paper No. 97-GT-115, (1997).
10. Zienkewicz, O. C. The Finite Element Method. 4th ed. McGraw-Hill Book Company, London, (1991).
11. Irretier, H. and Balashov, D. B. Transient response oscillations of a slow-variant systems with small non-linear damping: Modelling and prediction. J. of Sound and Vibration, 231(5), 1271-1287, (2000).
12. Irretier, H. and Leul, F. Non-stationary vibrations of mechanical systems with slowly varying natural frequencies during acceleration through resonance. Proceedings of the 9th World Congress on the Theory of Machine and Mechanisms, 1319-1323, (1995).
13. Asfar, K. R. Quenching of self-excited vibration. J. of Vibration, Acoustics, Stress, and Reliability in Design, 111(2), 130-133, (1989).
14. Takayama, K. A. Class of solvable second order ordinary differential equations with variable coefficients. J. of Mathematical Physics, 27(4), 1747-1762, (1984).
15. Antone, T. A., Al-Maaitah, A. A. Analytical solutions to classes of linear oscillator equations with time varying frequencies. J. Mathematical Physics, 33(10), 3330-3339, (1992).
16. Coddington, E. A. and Levinson, N. Theory of Ordinary Differential Equations. McGraw-Hill, New York, (1955).
Ayman A. Al-Maaitah , PhD, Associated Professor at Mutah University, Mutah, Jordan