A time-varying stiffness rotor-active magnetic bearings system under parametric excitation

2008 ◽  
Vol 222 (3) ◽  
pp. 447-458 ◽  
Author(s):  
U H Hegazy ◽  
Y A Amer
Keyword(s):  
Parametric Resonance ◽  
Parametric Excitation ◽  
Multiple Scales ◽  
Magnetic Bearings ◽  
Time Varying ◽  
Active Magnetic Bearings ◽  
Principal Parametric Resonance ◽  
Non Linear ◽  
Response Function Method ◽  
Non Linear System

The method of multiple scales is applied to investigate the non-linear oscillations and dynamic behaviour of a rotor-active magnetic bearings (AMBs) system, with time-varying stiffness. The rotor-AMB model is a two-degree-of-freedom non-linear system with quadratic and cubic non-linearities and parametric excitation in the horizontal and vertical directions. The case of principal parametric resonance is considered and examined. The steady-state response and the stability of the system at the principal parametric resonance case for various parameters are studied numerically, applying the frequency response function method. It is shown that the system exhibits many typical non-linear behaviours including multiple-valued solutions, jump phenomenon, hardening and softening non-linearity. Different effects of the system parameters on the non-linear response of the rotor are also reported. Results are compared with available published work.

A Time-Varying Stiffness Rotor Active Magnetic Bearings Under Combined Resonance

2008 ◽  
Vol 75 (1) ◽  
Author(s):  
U. H. Hegazy ◽  
M. H. Eissa ◽  
Y. A. Amer
Keyword(s):  
Multiple Scales ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Multiple Scale ◽  
Magnetic Bearings ◽  
Time Varying ◽  
Active Magnetic Bearings ◽  
Gyroscopic Effects ◽  
Nonlinear Equations Of Motion ◽  
Combined Resonance

This paper is concerned with the nonlinear oscillations and dynamic behavior of a rigid disk-rotor supported by active magnetic bearings (AMB), without gyroscopic effects. The nonlinear equations of motion are derived considering a periodically time-varying stiffness. The method of multiple scales is applied to obtain four first-order differential equations that describe the modulation of the amplitudes and the phases of the vibrations in the horizontal and vertical directions. The stability and the steady-state response of the system at a combination resonance for various parameters are studied numerically, applying the frequency response function method. It is shown that the system exhibits many typical nonlinear behaviors, including multiple-valued solutions, jump phenomenon, hardening, and softening nonlinearity. A numerical simulation using a fourth-order Runge-Kutta algorithm is carried out, where different effects of the system parameters on the nonlinear response of the rotor are reported and compared to the results from the multiple scale analysis. Results are compared to available published work.

Nonlinear Study of Rotor-AMB System Subject to Multi-Parametric Excitations

2013 ◽  
Vol 397-400 ◽  
pp. 359-364
Author(s):  
Lin Li ◽  
Yong Jie Han ◽  
Zheng Yi Ren
Keyword(s):  
Steady State ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Active Magnetic Bearings ◽  
Jump Phenomenon ◽  
State Response ◽  
Non Linear ◽  
Response Function Method ◽  
Amb System ◽  
Steady State Solutions

The non-linear response of a rotor supported by active magnetic bearings (AMB) under multi-parametric excitations is investigated. The method of multiple scales is applied to analyze the response of two modes of a rotorAMB system near the primary resonance case. The steady-state response of the system is studied numerically, applying the frequency response function method. It is shown that the system exhibits many typical non-linear behaviors including multiple-valued solutions, jump phenomenon, hardening non-linearity. The effects of the different parameters on the steady state solutions are investigated and discussed.

BIFURCATION OF MULTIPLE LIMIT CYCLES FOR A ROTOR-ACTIVE MAGNETIC BEARINGS SYSTEM WITH TIME-VARYING STIFFNESS

2008 ◽  
Vol 18 (03) ◽  
pp. 755-778 ◽  
Author(s):  
J. LI ◽  
Y. TIAN ◽  
W. ZHANG ◽  
S. F. MIAO
Keyword(s):  
Nonlinear Equation ◽  
Limit Cycles ◽  
Multiple Scales ◽  
Equation Of Motion ◽  
Magnetic Bearings ◽  
Time Varying ◽  
Active Magnetic Bearings ◽  
Detection Function ◽  
Single Degree Of Freedom ◽  
Amb System

The bifurcations of multiple limit cycles for a rotor-active magnetic bearings (AMB) system with the time-varying stiffness are considered in this paper. The governing nonlinear equation of motion is established for the rotor-AMB system with single-degree-of-freedom and parametric excitation. Using the method of multiple scales, the governing nonlinear equation of motion is first transformed to the averaged equation, which is in the form of a Z2-symmetric perturbed polynomial Hamiltonian system of degree 5. Then, the bifurcation theory of planar dynamical system and the method of detection function are utilized to analyze the bifurcations of multiple limit cycles of the averaged equation. Four groups of parametric controlling conditions are given to obtain the configurations of compound eyes. It is found that there exist respectively at least 17, 19, 21 and 22 limit cycles in the rotor-AMB system with the time-varying stiffness under the different controlling conditions.

Principal parametric resonance of axially accelerating viscoelastic strings with an integral constitutive law

2005 ◽  
Vol 461 (2061) ◽  
pp. 2701-2720 ◽  
Author(s):  
Li-Qun Chen
Keyword(s):  
Steady State ◽  
Parametric Resonance ◽  
Multiple Scales ◽  
Constitutive Law ◽  
Transverse Motion ◽  
Viscoelastic Parameters ◽  
Principal Parametric Resonance ◽  
Linearized Stability ◽  
Speed Fluctuation ◽  
Axial Speed

The steady-state transverse responses and the stability of an axially accelerating viscoelastic string are investigated. The governing equation is derived from the Eulerian equation of motion of a continuum, which leads to the Mote model for transverse motion. The Kirchhoff model is derived from the Mote model by replacing the tension with the averaged tension over the string. The method of multiple scales is applied to the two models in the case of principal parametric resonance. Closed-form expressions of the amplitudes and the existence conditions of steady-state periodical responses are presented. The Lyapunov linearized stability theory is employed to demonstrate that the first (second) non-trivial steady-state response is always stable (unstable). Numerical calculations show that the two models are qualitatively the same, but quantitatively different. Numerical results are also presented to highlight the effects of the mean axial speed, the axial-speed fluctuation amplitude, and the viscoelastic parameters.

Asymptotic trends in time-varying oscillatory period for a dual-staged torsional system

2016 ◽  
Vol 231 (22) ◽  
pp. 4126-4138
Author(s):  
Michael D Krak ◽  
Rajendra Singh
Keyword(s):  
Linear System ◽  
Mechanical System ◽  
Time Domain ◽  
Effective Stiffness ◽  
Step Response ◽  
Analysis Tool ◽  
Time Varying ◽  
Non Linear ◽  
Non Linear System ◽  
Oscillatory Period

The primary goal of this article is to propose a new analysis tool that estimates the asymptotic trends in the time-varying oscillatory period of a non-linear mechanical system. The scope is limited to the step-response of a torsional oscillator containing a dry friction element and dual-staged spring. Prior work on the stochastic linearization techniques is extended and modified for application in time domain. Subsequently, an instantaneous expected value operator and the concept of instantaneous effective stiffness are proposed. The non-linear system is approximated at some instant during the step-response by a linear time-invariant mechanical system that utilizes the instantaneous effective stiffness concept. The oscillatory period of the non-linear step-response at that instant is then approximated by the natural period of the corresponding linear system. The proposed method is rigorously illustrated via two computational example cases (a near backlash and near pre-load non-linearities), and the necessary digital signal processing parameters for time domain analysis are investigated. Finally, the feasibility and applicability of the proposed method is demonstrated by estimating the softening and hardening trends in the time-varying oscillatory period of the measured response for two laboratory experiments that contain clearance elements and multi-staged torsional springs.

Input-to-state stabilisation for a general non-linear system with time-varying input delay

2017 ◽  
Vol 11 (16) ◽  
pp. 2793-2800 ◽  
Author(s):  
Xiushan Cai ◽  
Lingxin Meng ◽  
Cong Lin ◽  
Wei Zhang ◽  
Leibo Liu
Keyword(s):  
Linear System ◽  
Input Delay ◽  
Time Varying ◽  
Non Linear ◽  
Non Linear System

Principal Parametric Resonance of Axially Accelerating Viscoelastic Strings

2005 ◽  
Author(s):  
Li-Qun Chen
Keyword(s):  
Steady State ◽  
Parametric Resonance ◽  
Multiple Scales ◽  
Transverse Motion ◽  
Viscoelastic Parameters ◽  
Principal Parametric Resonance ◽  
Linearized Stability ◽  
Speed Fluctuation ◽  
The Stability ◽  
Axial Speed

The steady-state transverse responses and the stability of an axially accelerating viscoelastic string are investigated. The governing equation is derived from the Eulerian equation of motion of a continuum, which leads to the Mote model for transverse motion. The Kirchhoff model is derived from the Mote model by replacing the tension with the averaged tension over the string. The method of multiple scales is applied to the two models in the case of principal parametric resonance. Closed-form expressions of the amplitudes and the existence conditions of steady-state periodical responses are presented. The Lyapunov linearized stability theory is employed to demonstrate that the first (second) nontrivial steady-state response is always stable (unstable). Numerical calculations show that the two models are qualitatively the same, but quantitatively different. Numerical results are also presented to highlight the effects of the mean axial speed, the axial speed fluctuation amplitude, and the viscoelastic parameters.

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