Stochastic stability of gyroscopic systems under bounded noise excitation. (English) Zbl 1535.70018
Summary: Dynamic stochastic stability of a two-degree-of-freedom gyroscopic system under bounded noise parametric excitation is studied in this paper through moment Lyapunov exponent and the largest Lyapunov exponent. A rotating shaft subject to stochastically fluctuating thrust is taken as a typical example. To obtain these two exponents, the gyroscopic differential equation of motion is first decoupled into Itô stochastic differential equations by using the method of stochastic averaging. Then mathematical transformations are used in these Itô equation to obtain a partial differential eigenvalue problem governing moment Lyapunov exponents, the slope of which at the origin is equal to the largest Lyapunov exponent. Depending upon the numerical relationship between the natural frequency and the excitation frequencies, the gyroscopic system may fall into four types of parametric resonance, i.e. no resonance, subharmonic resonance, combination additive resonance, and combination differential resonance.
The effects of noise and frequency detuning parameters on the parametric resonance are investigated. The results pave the way to utilize or control the vibration of gyroscopic systems under stochastic excitation.
References:
| [1] | Zheng, Z. L., Lu, F. M, He, X. T., Sun, J. Y., Xie, C. X. and He, C., Large displacement analysis of rectangular orthotropic membranes under stochastic impact loading, Int. J. Str. Stab. Dyn.16 (1) (2016) 1640007. · Zbl 1359.74257 |
| [2] | Merkin, D. R., Gyroscopic Systems (State Publishing House, Moscow, 1956) (in Russian). |
| [3] | Mettler, E., Stability and vibration problems of mechanical systems under harmonic excitation, in Dynamic Stability of Structures, ed. Herrmann, G. (Pergamon Press, New York, 1966), pp. 169-188. · Zbl 0153.27001 |
| [4] | Chetaev, N. G., The Stability of Motion (Pergamon Press, New York, 1961). |
| [5] | Dimentberg, F. M., Flexural Vibrations of Rotating Shafts (Butterworths, London, 1961). |
| [6] | Huseyin, K., Vibrations and Stability of Multiple Parameter Systems (Noordhoff International Publishing, The Netherlands, 1978). · Zbl 0399.73032 |
| [7] | Ariaratnam, S. T. and Namachchivaya, N. Sri, Periodically perturbed linear gyroscopic systems, J. Str. Mech.14 (2) (1986) 127-151. |
| [8] | Ariaratnam, S. T. and Namachchivaya, N. Sri, Periodically perturbed nonlinear gyroscopic systems, J. Str. Mech.14 (2) (1986) 153-175. |
| [9] | Parszewski, Z. A., Principles of rotor system instability, in Dynamics of Rotors—Stability and Stiffness Identification, ed. Mahrenholtz, O. (Springer-Verlag, New York, 1984), pp. 65-88. |
| [10] | Namachchivaya, N. Sri and Ariaratnam, S. T., Stochastically perturbed linear gyroscopic systems, Mech. of Str. Mach.15 (3) (1987) 323-345. · Zbl 0635.34037 |
| [11] | Nagata, W. and Namachchivaya, N. Sri, Bifurcations in gyroscopic systems with an application to rotating shafts, Proc. Roy. Soc. A: Math. Phy. Eng. Sci.454 (1998) 543-585. · Zbl 0913.70008 |
| [12] | N. L. Vedula, Dynamics and stability of parametrically excited gyroscopic systems, PhD thesis, University of Illinois at Urbana-Champaign (2005). |
| [13] | Deng, J., Dynamic stability of a viscoelastic rotating shaft under parametric random excitation, Int. J. Non-Linear Mech.84 (2016) 56-64. |
| [14] | N. M. Abdelrahman, Stochastic stability of viscoelastic dynamical systems, PhD Thesis, University of Waterloo, Waterloo, Canada (2002). |
| [15] | Baxendale, P. and Stroock, D., Large deviations and stochastic flows of diffeomorphisms, Prob. Theo. Rel. Fields80 (1988) 169-215. · Zbl 0638.60035 |
| [16] | Arnold, L., Doyle, N. M. and Namachchivaya, N. Sri, Small noise expansion of moment Lyapunov exponents for two-dimensional systems, Dyn. Stab. Sys.12 (3) (1997) 187-122. · Zbl 0890.60055 |
| [17] | Khasminskii, R. Z., Stochastic Stability of Differential Equations (Kluwer Academic Publishers, Norwell, MA, 1980). · Zbl 1259.60058 |
| [18] | Xie, W.-C., Dynamic Stability of Structures (Cambridge University Press, Cambridge, 2006). · Zbl 1116.70001 |
| [19] | Namachchivaya, N. Sri and Van Roessel, H. J., Moment Lyapunov exponent and stochastic stability of two coupled oscillators driven by real noise, J. Appl. Mech.68 (6) (2001) 903-914. · Zbl 1110.74599 |
| [20] | Kozic, P., Janevski, G. and Pavlovic, R., Moment Lyapunov exponents and stochastic stability for two coupled oscillators, J. Mech. Mat. Str.4 (10) (2009) 1689-1701. |
| [21] | Stojanovic, V. and Petkovic, M., Moment Lyapunov exponents and stochastic stability of a three-dimensional system on elastic foundation using a perturbation approach, J. Appl. Mech.80 (5) (2013) 051009. |
| [22] | McDonald, R. J. and Namachchivaya, N. Sri, Parametric stabilization of a gyroscopic system, J. Sound Vib.255 (4) (2002) 635-662. |
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