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In this study, the critical conditions for generating chaos in a Duffing oscillator with nonlinear damping and fractional derivative are investigated. The Melnikov function of the Duffing oscillator is established based on Melnikov theory. The necessary analytical conditions and critical value curves of chaotic motion in the sense of Smale horseshoe are obtained. The numerical solutions of chaotic motion, including time history diagram, frequency spectrum diagram, phase diagram, and Poincare map, are studied. The correctness of the analytical solution is verified through a comparison of numerical and analytical calculations. The effects of linear and nonlinear parameters on chaotic motion are also analyzed. These results are relevant to the study of system dynamics.

Highly nonlinear vibration is one of important factors influencing the failure of gas foil bearing-rotor systems. In this paper, an integrated nonlinear dynamic model is constructed to study the dynamic vibration of worn gas foil bearings. In the wear model, the deformation of foil is considered. The Iordanoff model is used to solve the stiffness of foil, and equivalent viscous damping is introduced to amend the friction factor at the interface between the foil and bearing bracket. The incompressible Reynolds equation is solved by using ADI method, and the Newmark-β method is applied to obtain the trajectory of axis center. In numerical examples, the effects of worn zone on the trajectory of axis center, the Poincare map and bifurcation diagram are discussed. The nonlinear vibration resulting from the worn zone is revealed. Comparison with existing results validates this model. [ABSTRACT FROM AUTHOR]

Based on mixed thermal-elastohydrodynamic lubrication, the dynamic contact of ball bearings with worn band is investigated, and the film thickness, pressure, profile and interfacial temperature are analyzed. The temperature-pressure-viscoelastic equation of lubricant is constructed. The surface roughness, the elastic deformation and the worn band are included in the film thickness. The discretization method and compound iteration method are combined to solve the problem. Through numerical examples, the distribution and contour of film thickness, pressure and temperature with consideration of surface roughness and worn band are discussed, and the point contact with lower entrainment speed is found. [ABSTRACT FROM AUTHOR]

A new kind of three-element type dynamic vibration absorber (DVA) with negative-stiffness spring is studied in detail. At first, the analytical solution of the system is obtained based on the established motion differential equation. Three fixed points are found in the amplitude-frequency curves of the primary system. The design formulae for the optimal tuning ratio and optimal stiffness ratio of the DVA are obtained by adjusting the three fixed points to the same height according to the fixed-point theory. According to the characteristics of negative-stiffness element, the optimal negative stiffness ratio is obtained and it could keep the system stable. Then the optimal damping ratio is obtained by minimizing the maximum value of the amplitude-frequency curves. The comparison between the analytical solution and the numerical one verifies the correctness of the analytical solution. The comparisons the presented DVA with three other traditional DVAs under the harmonic and random excitations show that the presented DVA in this paper performs better in vibration absorption. This result could provide theoretical basis for the optimal design of similar DVAs. [ABSTRACT FROM AUTHOR]