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

2021 ◽  
Vol 26 (2) ◽  
pp. 33
Author(s):  
Muhammad Usman ◽  
Shaaban Abdallah ◽  
Mudassar Imran
Keyword(s):  
Stability Analysis ◽  
Perturbation Method ◽  
Primary Resonance ◽  
Bifurcation Curve ◽  
Stability Theorems ◽  
Asymptotic Perturbation Method ◽  
Restoring Forces ◽  
The Stability ◽  
External Forces ◽  
Asymptotic Perturbation

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.

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

2017 ◽  
Author(s):  
Ruiqin Wu ◽  
Wei Zhang ◽  
Ming Hui Yao
Keyword(s):  
Nonlinear Dynamics ◽  
Perturbation Method ◽  
Primary Resonance ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Chaotic Motions ◽  
Dimensionless Equation ◽  
Asymptotic Perturbation Method ◽  
Amb System ◽  
Asymptotic Perturbation

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.

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

2020 ◽  
Vol 2020 ◽  
pp. 1-29 ◽  
Author(s):  
W. Zhang ◽  
R. Q. Wu ◽  
B. Siriguleng
Keyword(s):  
Perturbation Method ◽  
Electromagnetic Force ◽  
Nonlinear Vibrations ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Time Varying ◽  
Asymptotic Perturbation Method ◽  
Amb System ◽  
Asymptotic Perturbation ◽  
Quadratic And Cubic Nonlinearities

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.

scholarly journals Stability Analysis for Autonomous Dynamical Switched Systems through Nonconventional Lyapunov Functions

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
V. Nosov ◽  
J. A. Meda-Campaña ◽  
J. C. Gomez-Mancilla ◽  
J. O. Escobedo-Alva ◽  
R. G. Hernández-García
Keyword(s):  
Stability Analysis ◽  
Finite Number ◽  
Switched Systems ◽  
Lyapunov Functions ◽  
Switched System ◽  
Linear Functions ◽  
Asymptotically Stable ◽  
Multiple Lyapunov Functions ◽  
Stability Theorems ◽  
The Stability

The stability of autonomous dynamical switched systems is analyzed by means of multiple Lyapunov functions. The stability theorems given in this paper have finite number of conditions to check. It is shown that linear functions can be used as Lyapunov functions. An example of an exponentially asymptotically stable switched system formed by four unstable systems is also given.

Partial stability analysis of nonlinear time-varying impulsive systems

2019 ◽  
Vol 12 (06) ◽  
pp. 1950066
Author(s):  
Boulbaba Ghanmi
Keyword(s):  
Stability Analysis ◽  
Lyapunov Functions ◽  
Time Varying ◽  
Impulsive Systems ◽  
Partial Stability ◽  
Stability Theorems ◽  
Time Varying Systems ◽  
The Stability ◽  
Derivatives Of ◽  
Theoretical Results

This paper investigates the stability analysis with respect to part of the variables of nonlinear time-varying systems with impulse effect. The approach presented is based on the specially introduced piecewise continuous Lyapunov functions. The Lyapunov stability theorems with respect to part of the variables are generalized in the sense that the time derivatives of the Lyapunov functions are allowed to be indefinite. With the help of the notion of stable functions, asymptotic partial stability, exponential partial stability, input-to-state partial stability (ISPS) and integral input-to-state partial stability (iISPS) are considered. Three numerical examples are provided to illustrate the effectiveness of the proposed theoretical results.

Lyapunov-Based Sufficient Conditions for Nonlinear Impulsive Systems

2011 ◽  
Vol 219-220 ◽  
pp. 508-512
Author(s):  
Yong Liang Gao ◽  
Xiao Wu Mu
Keyword(s):  
Differential Equation ◽  
Stability Analysis ◽  
Lyapunov Function ◽  
Sufficient Conditions ◽  
Invariant Set ◽  
Positive Definite ◽  
Impulsive Systems ◽  
Stability Theorems ◽  
Sufficient Conditions For Convergence ◽  
The Stability

This paper focuses on the stability analysis and invariant set stability theorems for nonlinear impulsive systems. A set of Lyapunov-based sufficient conditions are established for these convergent properties. These results do not require the Lyapunov function to be positive definite. Inequalities relating the righthandside of the differential equation and the Lyapunov function derivative are involved for these results. These inequalities make it possible to deduce properties of the functions and thus leads to sufficient conditions for convergence and stability.

scholarly journals Stability Analysis and Reinforcement of a High-Steep Rock Slope with Faults: Numerical Analysis and Field Monitoring

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Qibing Zhan ◽  
Xinjian Sun ◽  
Cheng Li ◽  
Yawei Zhao ◽  
Xinjie Zhou ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Numerical Analysis ◽  
Rock Slope ◽  
Field Monitoring ◽  
Monitoring Data ◽  
Slope Surface ◽  
Monitoring Point ◽  
Reinforcement Measure ◽  
The Stability ◽  
External Forces

This study presents a stability analysis of a high-steep rock slope with two faults during excavations and evaluates the effectiveness of a proposed reinforcement method using prestressed anchor cables. A 3D finite difference model was established based on the strength reduction method using FLAC3D software. The influence of various fault conditions and the effectiveness of the reinforcement on the slope stability during the excavation process were analyzed and compared to field monitoring data. The numerical analysis and field monitoring results showed that the fault close to the slope surface (f20) was prone to the local instability under external forces caused by the excavation, but a fault further away from the slope surface (f14) had insignificant influence on the stability of the slope. Based on the numerical analysis results, the proposed reinforcement measure can increase the factor of safety (FOS) of the slope by 19.2%. The field monitoring data also showed that the displacement of the monitoring point gradually decreased after the reinforcement, and the deformation of the slope was effectively controlled.

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

2008 ◽  
Vol 47-50 ◽  
pp. 1137-1140
Author(s):  
Yu Xin Hao ◽  
Wei Zhang ◽  
Jie Yang ◽  
Li Hua Chen
Keyword(s):  
Perturbation Method ◽  
Degrees Of Freedom ◽  
Functionally Graded ◽  
Bottom Surface ◽  
Graded Materials ◽  
Chaotic Motions ◽  
Simply Supported ◽  
Asymptotic Perturbation Method ◽  
Asymptotic Perturbation ◽  
Two Degree 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.

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