Nonlinear and Stability Analysis of a Ship with General Roll-Damping Using an Asymptotic Perturbation Method

In this work, the response of a ship rolling in regular beam waves is studied. The model is one degree of freedom model for nonlinear ship dynamics. The model consists of the terms containing inertia, damping, restoring forces, and external forces. The asymptotic perturbation method is used to study the primary resonance phenomena. The effects of various parameters are studied on the stability of steady states. It is shown that the variation of bifurcation parameters affects the bending of the bifurcation curve. The slope stability theorems are also presented.

Nonlinear Vibrations of a Rotor-Active Magnetic Bearing System with 16-Pole Legs and Two Degrees of Freedom

The asymptotic perturbation method is used to analyze the nonlinear vibrations and chaotic dynamics of a rotor-active magnetic bearing (AMB) system with 16-pole legs and the time-varying stiffness. Based on the expressions of the electromagnetic force resultants, the influences of some parameters, such as the cross-sectional area Aα of one electromagnet and the number N of windings in each electromagnet coil, on the electromagnetic force resultants are considered for the rotor-AMB system with 16-pole legs. Based on the Newton law, the governing equation of motion for the rotor-AMB system with 16-pole legs is obtained and expressed as a two-degree-of-freedom system with the parametric excitation and the quadratic and cubic nonlinearities. According to the asymptotic perturbation method, the four-dimensional averaged equation of the rotor-AMB system is derived under the case of 1 : 1 internal resonance and 1 : 2 subharmonic resonances. Then, the frequency-response curves are employed to study the steady-state solutions of the modal amplitudes. From the analysis of the frequency responses, both the hardening-type nonlinearity and the softening-type nonlinearity are observed in the rotor-AMB system. Based on the numerical solutions of the averaged equation, the changed procedure of the nonlinear dynamic behaviors of the rotor-AMB system with the control parameter is described by the bifurcation diagram. From the numerical simulations, the periodic, quasiperiodic, and chaotic motions are observed in the rotor-active magnetic bearing (AMB) system with 16-pole legs, the time-varying stiffness, and the quadratic and cubic nonlinearities.

Analysis of Nonlinear Dynamics of a Rotor-Active Magnetic Bearing System With 16-Pole Legs

In this paper, we use the asymptotic perturbation method to analyze the nonlinear dynamics of a rotor-active magnetic bearing (AMB) system with 16-pole legs. The motion governing equation is derived by using classical Newton law. The resulting dimensionless equation of motion for the system is expressed as a two-degree-of-freedom system including the parametric excitation, quadratic and cubic nonlinearities. The asymptotic perturbation method is used to obtain the averaged equation when the primary resonance and 1/2 sub-harmonic resonance are taken into consideration. From the averaged equations obtained, numerical simulations are presented to investigate the modulation of vibration amplitudes of the rotor-AMB system. Based on a specific set of parameters, it is found that there exist the periodic, quasi-periodic and chaotic motions in the modulated amplitude of the rotor in the system.

Nonlinear Oscillation of Shell Workpiece in High Speed Milling Under 1:2 Internal Resonance Condition

We aim to study nonlinear dynamics of a shell-shaped workpiece during milling processes in this paper. The shell-shaped workpiece is modelled as a cantilever thin shell subjected to a cutting force with time-delay effects. The formulas of the cantilever shell were derived by the classical shell theory and the von Karman strain-displacement relations. The resulting differential equations are reduced to a two-degree-of-freedom nonlinear system ordinary differential equations by applying the Galerkin’s approach. The method of Asymptotic Perturbation method is used to obtain the averaged equations, which were dealt with the resonance cases of 1:2 internal resonance and principal parametric resonance. Dynamic behaviors are presented based on the numerical solutions. The results show that different time-delay parameters result in periodic motion, multiple periodic motion, and chaotic motion.

Periodic and Chaotic Motions of a Simply Supported FGM Plate with Two Degrees-of-Freedom

In this paper, we use the asymptotic perturbation method to investigate the nonlinear oscillation and chaotic dynamic behavior of a simply supported rectangular plate made of functionally graded materials (FGMs). We assume that the plate is made from a mixture of ceramics and metals with continuously varying compositional profile such that the top surface of the plate is ceramic rich, whereas the bottom surface is metal rich. The equations motion of the FGM plate with two-degree-of-freedom under combined parametrical and external excitations are obtained by using Galerkin’s method. Based on the averaged equation obtained by the asymptotic perturbation method, the phase portrait and waveform are used to analyze the periodic and chaotic motions. It is found that the FGM plate exhibits chaotic motions under certain circumstances.

Multi-Pulse Chaotic Motions of a Rotor-Active Magnetic Bearing With Time-Varying Stiffness

In this paper, we investigate the Shilnikov type multi-pulse chaotic dynamics for a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time varying in a periodic form. The dimensionless equation of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions is a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found from the numerical results that there are the phenomena of the Shilnikov type multi-pulse chaotic motions for the rotor-AMB system. A new jumping phenomenon is discovered in the rotor-AMB system with the time-varying stiffness.

A model equation for non-resonant interacting ion acoustic plasma waves

The interaction among non-resonant ion acoustic plasma waves with different group velocities that are not close to each other is studied by an asymptotic perturbation method, based on Fourier expansion and spatio-temporal rescaling. It is shown that the nonlinear Schrödinger equation is not adequate, and instead a model system of nonlinear evolution equations is necessary to describe oscillation amplitudes of Fourier modes. This system is C-integrable, i.e. it can be linearized through an appropriate transformation of the dependent and independent variables. We demonstrate that the subclass of localized solutions gives rise to a solitonic phenomenology. These solutions propagate with the relative group velocity and maintain their shape during a collision, the only change being a phase shift. Numerical calculations confirm the validity of these predictions.