The oil-film oscillation in a large rotating machinery is a complex high-dimensional nonlinear problem. In this paper, a high pressure rotor of an aero engine with a pair of liquid-film lubricated bearings is modeled as a twenty-two-degree-of-freedom nonlinear system by the Lagrange method. This high-dimensional nonlinear system can be reduced to a two-degree-of-freedom system preserving the oil-film oscillation property by introducing the modified proper orthogonal decomposition (POD) method. The efficiency of the method is shown by numerical simulations for both the original and reduced systems. The Chen-Longford (C-L) method is introduced to get the dynamical behaviors of the reduced system that reflect the natural property of the oil-film oscillation.
In this paper, the periodic solutions of the smooth and discontinuous (SD) oscillator, which is a strongly irrational nonlinear system are discussed for the system having a viscous damping and an external harmonic excitation. A four dimensional averaging method is employed by using the complete Jacobian elliptic integrals directly to obtain the perturbed primary responses which bifurcate from both the hyperbolic saddle and the non-hyperbolic centres of the unperturbed system. The stability of these periodic solutions is analysed by examining the four dimensional averaged equation using Lyapunov method. The results presented herein this paper are valid for both smooth ( α > 0) and discontinuous ( α = 0) stages providing the answer to the question why the averaging theorem spectacularly fails for the case of medium strength of external forcing in the Duffing system analysed by Holmes. Numerical calculations show a good agreement with the theoretical predictions and an excellent efficiency of the analysis for this particular system, which also suggests the analysis is applicable to strongly nonlinear systems.
In this paper,we propose a novel nonlinear oscillator with strong irrational nonlinearities having smooth and discontinuous characteristics depending on the values of a smoothness parameter.The oscillator is similar to the SD oscillator,originally introduced in Phys Rev E 69(2006).The equilibrium stability and the complex bifurcations of the unperturbed system are investigated.The bifurcation sets of the equilibria in parameter space are constructed to demonstrate transitions in the multiple well dynamics for both smooth and discontinuous regimes.The Melnikov method is employed to obtain the analytical criteria of chaotic thresholds for the singular closed orbits of homoclinic,homo-heteroclinic,cuspidal heteroclinic and tangent homoclinic orbits of the perturbed system.
Nonlinear dynamical systems with an irrational restoring force often occur in both science and engineering,and always lead to a barrier for conventional nonlinear techniques.In this paper,we have investigated the global bifurcations and the chaos directly for a nonlinear system with irrational nonlinearity avoiding the conventional Taylor's expansion to retain the natural characteristics of the system.A series of transformations are proposed to convert the homoclinic orbits of the unperturbed system to the heteroclinic orbits in the new coordinate,which can be transformed back to the analytical expressions of the homoclinic orbits.Melnikov's method is employed to obtain the criteria for chaotic motion,which implies that the existence of homoclinic orbits to chaos arose from the breaking of homoclinic orbits under the perturbation of damping and external forcing.The efficiency of the criteria for chaotic motion obtained in this paper is verified via bifurcation diagrams,Lyapunov exponents,and numerical simulations.It is worthwhile noting that our study is an attempt to make a step toward the solution of the problem proposed by Cao Q J et al.(Cao Q J,Wiercigroch M,Pavlovskaia E E,Thompson J M T and Grebogi C 2008 Phil.Trans.R.Soc.A 366 635).
