Bifurcation and Erosion of Safe Basin for a Spur Gear System

Tooth surface contact deformation is part of the main causes for gear system vibration. The safety region of the single-stage spur system vibration is established based on the tooth surface contact deformation. Safe basin and its erosion are calculated numerically according to the safety region as the system control parameters are varied. The basin of attraction in the safe basin is computed by combining the simple cell-to-cell method. Vibration safety and global dynamics of the gear system are investigated. Some bifurcations and mechanisms of the erosion of safe basin are studied by using phase portraits, Poincaré maps, bifurcation diagrams under multi-initial values and top Lyapunov exponents (TLE). The sensitivity of system behavior and bifurcation to initial values are discussed as well. Hidden bifurcating points and attractors are revealed. To get a better understanding of the sensitivity of the system behavior to initial values, a bifurcation dendrogram under multi-initial values is designed. It is found that there is the erosion of safe basin existing in the examining area. Both vibration amplitude changing of coexisting attractors and appearing or disappearing of coexisting attractors are the main causes for the erosion of the safe basin. Bifurcation of the system behavior is selective to initial values. With varying frequency, backlash and comprehensive transmission error, the period-1 response gradually bifurcates under multi-initial conditions and evolves into other new periodic responses that coexist with the period-1 response. The new coexisting response has a key effect on the erosion of the safe basin. The study is helpful to the optimization design of the gear system and the control of the system behavior.

Stochastic Resonance and Safe Basin of Single-Walled Carbon Nanotubes with Strongly Nonlinear Stiffness under Random Magnetic Field

In this paper, a kind of single-walled carbon nanotube nonlinear model is developed, and the strongly nonlinear dynamic characteristics of such carbon nanotubes subjected to random magnetic field are studied. The nonlocal effect of microstructure is considered based on the theory of nonlocal elasticity. The natural frequency of the strongly nonlinear dynamic system is obtained by the energy function method, the drift coefficient and the diffusion coefficient are verified. The stationary probability density function of the system dynamic response is given and the fractal boundary of the safe basin is provided. Theoretical analysis and numerical simulation show that stochastic resonance occurs when varying the random magnetic field intensity. The boundary of safe basin has fractal characteristics and the area of safe basin decreases when the intensity of the magnetic field permeability increases.

Safe basins for a nonlinear oscillator with ramped forcing

A nonlinear oscillator is studied in the presence of external forcing for which the amplitude initially depends on time. The focus is on the sizes of the basins of attraction which do not lead to unbounded motions, collectively termed the ‘safe basin’. Direct comparisons are drawn between the regime of constant forcing amplitude and that where the forcing amplitude initially depends on time. In the process, questions from previous literature are answered and previously unexplained phenomena are understood. Furthermore, we witness a new phenomenon, not previously observed for the system studied.

Numerical Method Research on Nonlinear Roll System of Large Container Ship

When dealing with the ship-roll problem, the roll motion is mainly regarded as a single degree-of-freedom dynamical system, and the nonlinear properties are reflected in the nonlinear damping term and restoring moment term. Previous studies have shown that transverse chaotic phenomenon means the damage of ship-roll stability which will lead to ship capsizing, and for ultra large container ships, the wind area above water surface can not be neglected, which turns the ship-roll system into asymmetric dynamical system. The concept of safe basin is usually used to express the boundedness of motion. It is defined as the set of bounded solution to dynamical system, and the erosion phenomenon of safe basin is normally explained as the global instability. This concept was firstly brought out by Thompson [1] when he studied the problem of ship capsizing and then was applied to different fields of engineering. Based on this background, the following three tasks are completed in this paper. a) For the calculation of chaos threshold, two numerical methods, namely, Pade approximation and Gauss-Legendre integration are adopted, analyzed and compared. b) One 9200TEU container ship is selected and the chaos threshold is calculated by virtue of Gauss-Legendre method. As numerical verification, the gradually erosion phenomenon of ship’s safe basin is observed and phase trajectories of points located in broken domain are traced; c) When encountered with crosswind (When winds are not parallel to or directly against the line of travel, the wind is said to have a crosswind component; that is, the force can be separated into two vector components, a crosswind component and a headwind or tailwind component.), the symmetry of ship-roll system begin to break. In the last part of this paper, the effect of crosswind on safe basin, asymmetry, and stability are studied.

Chaos Research of Asymmetric System Based on Melnikov Method

Based on Helmholtz-Duffing which is a typical nonlinear asymmetric dynamical system, According to homoclinic bifurcations, prerequisites for chaos motion are obtained by use of Melnikov theory. As a result of verifying the analytic solutions, the safe basin and the phenomena of erosion of the safe basin are simulated by numerical method in the end. The research indicates that the value range of is closely related to the area influenced by the value itself: when is located in the range of from zero to one, the left part of system is mainly affected; when is larger than one, the right part of system is mainly affected. For the same symmetry parameter , there exist a critical frequency at which the threshold value of the amplitudes of both left and right part of system are equal.