On Double Frequency Vibration of a Generator-Bearing System With Asymmetrical Stiffness

2002 ◽  
Author(s):  
Senlin Huang ◽  
Zhansheng Liu ◽  
Jiexian Su
Keyword(s):  
Degrees Of Freedom ◽  
Reduction Method ◽  
Integration Method ◽  
Element Model ◽  
Modal Interaction ◽  
Motion Equations ◽  
Double Frequency ◽  
Modal Reduction ◽  
Direct Integration Method ◽  
Bearing System

A finite element model for a generator-bearing system with asymmetrical stiffness is developed for investigation of the double frequency vibration. The modal reduction method is used for reducing the degrees of freedom system to improve computing efficiency, and the Newmark direct integration method is employed to solve the reduced motion equations. The two-modal interaction vibration is induced when the rotation speed is half a critical speed of the system due to asymmetry and gravity force of the generator. Such a phenomena is observed in the practical test.

scholarly journals Extension of modal reduction methods to non-linear coupled structure-acoustic problems

2011 ◽  
pp. 227-245
Author(s):  
Youssef Gerges ◽  
Emeline Sadoulet-Reboul ◽  
Morvan Ouisse ◽  
Noureddine Bouhaddi
Keyword(s):  
Numerical Simulations ◽  
Linear Model ◽  
Elastic Plate ◽  
Degrees Of Freedom ◽  
Reduction Method ◽  
Static Response ◽  
Acoustic Cavity ◽  
Modal Reduction ◽  
Non Linear ◽  
Reduction Methods

This paper proposes a robust reduction method dedicated to non-linear vibroacoustic problems in the context of localized geometrical non-linearities. The method consists in enriching the truncated uncoupled modal basis of the linear model by a static response due to unit forces on the non-linear degrees of freedom and by the static response of the fluid due to the interaction with the structure. To show the effectiveness of the proposed method, numerical simulations of responses of an elastic plate closing an acoustic cavity and a hang-on exhaust are performed.

Modal Reduction Method for Vibrational Analysis of Structures Partially Filled With Internal Liquids

2005 ◽  
Author(s):  
Jean-Se´bastien Schotte´ ◽  
Roger Ohayon
Keyword(s):  
Degrees Of Freedom ◽  
Reduction Method ◽  
Flexible Structures ◽  
Vibrational Analysis ◽  
Galerkin Projection ◽  
The Finite Element Method ◽  
Aircraft Wings ◽  
System A ◽  
Modal Reduction ◽  
Fluid Sloshing

In the framework of the vibrational analysis of structures, we propose a method to take into account the internal liquids (like fuel) in the structure modelling. Since the “classical” added-mass hydroelastic model cannot be used for slender and flexible structures such as space launchers or civil aircraft wings (because the decoupling assumption of sloshing and hydroelasticity is not valid for such structures), we propose here a revised hydroelastic formulation which takes this coupling into account. Since this model introduces in the system a great number of degrees of freedom for the fluid, we will present a reduction method using the Ritz-Galerkin projection onto the fluid sloshing modal basis. A discretization of the reduced equations by the finite element method will be proposed and the convergence rate of this modal reduction method will be discussed on an application example.

The Study of Explicit Scheme of Hybrid Stress-Function Element with Rotation Degrees of Freedom

2013 ◽  
Vol 353-356 ◽  
pp. 3220-3223
Author(s):  
Li Na Ge ◽  
Ge Tian ◽  
Ming Wu Yuan ◽  
Meng Yan Song ◽  
Xiang Rong Fu
Keyword(s):  
Degrees Of Freedom ◽  
Integration Method ◽  
Computational Cost ◽  
Element Model ◽  
Fundamental Solutions ◽  
Plane Elasticity ◽  
Explicit Scheme ◽  
Triangular Element ◽  
Triangular Area ◽  
Triangular Area Coordinates

A simple and efficient explicit scheme of triangular planar element with rotation degrees of freedom is proposed in this paper. The basic fundamental solutions of plane elasticity problem based on Airy stress functions are used as trial functions to construct triangular element with drilling degrees of freedom. During the construction of element model, the explicit expression of element stiffness matrix is deduced by means of triangular area coordinates integration method, instead of numerical integration method. Numerical calculation indicates that the element constructed in this paper is of high precision but less computational cost.

The Extended Modal Reduction Method Applied to Rotor Dynamic Problems

1991 ◽  
Vol 113 (1) ◽  
pp. 79-84 ◽  
Author(s):  
K. Kane ◽  
B. J. Torby
Keyword(s):  
Degrees Of Freedom ◽  
Reduction Method ◽  
Mode Shapes ◽  
Dynamic Problems ◽  
Modal Reduction ◽  
Stiffness Matrices ◽  
Space Formulation ◽  
Gyroscopic Effects ◽  
Stiffness And Damping ◽  
Rotor Dynamic

In this paper, the existing Modal Reduction Method, which was developed to handle symmetric mass and stiffness matrices, is extended utilizing state-space formulation to handle nonsymmetric mass, damping, and stiffness matrices. These type of matrices typically accompany rotor dynamic problems since journal bearings supporting the rotor have nonsymmetric stiffness and damping characteristics. The purpose of modal reduction is to eliminate unimportant modes and degrees of freedom from the analytical model after they are found, so that further numerical analysis can be accelerated. The reduction described here leaves the retained eigenvalues and mode shapes unaltered from their original values. This method is demonstrated for a simple rotor problem having nonsymmetric system matrices including gyroscopic effects.

Dynamic Behavior of a Mistuned Air Turbine: Comparison Between Simulations and Measurements

2014 ◽  
Author(s):  
Linus Pohle ◽  
Lars Panning-von Scheidt ◽  
Joerg Wallaschek ◽  
Jens Aschenbruck ◽  
Joerg R. Seume
Keyword(s):  
Degrees Of Freedom ◽  
Reduction Method ◽  
Turbine Blades ◽  
Maximum Amplitude ◽  
Element Model ◽  
Computational Effort ◽  
Air Turbine ◽  
Blade Row ◽  
Manufacturing Tolerances ◽  
The One

Due to manufacturing tolerances, wear during operation or regeneration processes like maintenance operation, the structural properties of turbine blades deviate from design condition to reference blades. This deviation usually causes higher vibration amplitudes and as a consequence a lower service life expectation. Many different calculation methods can be used to simulate these increased amplitudes of mistuned blades. The major resulting problem is on the one hand to capture the occurring deviation of the eigenfrequencies from the reference blade and on the other hand to incorporate these real deviations in simulations. Solving these problems with a simplified experimental setup will make it possible to predict the maximum amplitude and to avoid costly experiments in a rotating turbine. The aim of the paper is to verify a simulation of the vibration amplitude by experiments using a reduction method to calculate a mistuned system in reasonable time. The results of the chosen simulation are compared to experiments in a rotating turbine. To reduce the number of degrees of freedom of the full finite-element model and the computational effort, a multi-step reduction method is used. In the simulation, the centrifugal force, the structural damping, the steady static pressure on the blades, and the mistuning are considered. To find the occurring deviations of each manufactured blade, an experimental modal analysis is performed for every single blade in a non-rotating setup with the eigenfrequencies of every single blade as an output. The single-stage results of the simulation are subsequently compared to experiments in a 5-stage air turbine in which the vibration amplitudes and the eigenfrequencies of every blade in the last rotor blade row are measured by a tip-timing system.

Finite-Element-Based Nonlinear Modal Reduction of a Rotating Beam with Large-Amplitude Motion

2003 ◽  
Vol 9 (3-4) ◽  
pp. 235-263 ◽  
Author(s):  
Polarit Apiwattanalunggarn ◽  
Steven W. Shaw ◽  
Christophe Pierre ◽  
Dongying Jiang
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Degrees Of Freedom ◽  
Element Model ◽  
Reduced Order Model ◽  
Rotating Beam ◽  
Order Model ◽  
Reduced Order ◽  
Modal Reduction ◽  
The Finite Element Model

A nonlinear one-dimensional finite-element model representing the axial and transverse motions of a cantilevered rotating beam is reduced to a single nonlinear normal mode using invariant manifold techniques. This system is an idealized representation for large-amplitude vibrations of a rotorcraft blade. Although this model is relatively simple, it possesses the essential nonlinear coupling effects between the axial and transverse degrees of freedom. The nature of this coupling leads to the fact that we must use many degrees of freedom, whether based on finite elements or modal expansions, in order to accurately represent the beam vibrations. In this work, the slow modal convergence problem is overcome by nonlinear modal reduction that makes use of invariant manifold based nonlinear modes. This reduction procedure generates a single-degree-of-freedom reduced-order model that systematically accounts for the dynamics of all the linear modes, or finite elements, considered in the original model. The approach is used to study the dynamic characteristics of the finite-element model over a wide range of vibration amplitudes. Using extensive simulations, it is shown that the response of the reduced-order model is nearly identical to a reference system which is based on a large-scale representation of the finite-element model, and to a reduced-order Rayleigh-Ritz model. All of the procedures presented here have been computationally automated. Hence, in this study we demonstrate that it is feasible and practical to interface nonlinear finite-element methods with nonlinear modal reduction.

Bright and dark optical solitons for the generalized variable coefficients nonlinear Schrödinger equation

2020 ◽  
Vol 21 (7-8) ◽  
pp. 675-681
Author(s):  
Rehab M. El-Shiekh ◽  
Mahmoud Gaballah
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Integration Method ◽  
Variable Coefficients ◽  
Nonlinear Schrodinger Equation ◽  
Equation Method ◽  
Direct Integration ◽  
Wave Solutions ◽  
Direct Integration Method

AbstractIn this paper, the generalized nonlinear Schrödinger equation with variable coefficients (gvcNLSE) arising in optical fiber is solved by using two different techniques the trail equation method and direct integration method. Many different new types of wave solutions like Jacobi, periodic and soliton wave solutions are obtained. From this study we have concluded that the direct integration method is more easy and straightforward than the trail equation method. As an application in optic fibers the propagation of the frequency modulated optical soliton is discussed and we have deduced that it's propagation shape is affected with the different values of both the amplification increment and the group velocity (GVD).

scholarly journals Precise Integration Method for Solving Noncooperative LQ Differential Game

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Hai-Jun Peng ◽  
Sheng Zhang ◽  
Zhi-Gang Wu ◽  
Biao-Song Chen
Keyword(s):  
Differential Equation ◽  
Differential Game ◽  
Integration Method ◽  
Transformation Method ◽  
Linear Quadratic ◽  
Riccati Differential Equation ◽  
Precise Integration ◽  
Precise Integration Method ◽  
Direct Integration Method ◽  
Two Parameters

The key of solving the noncooperative linear quadratic (LQ) differential game is to solve the coupled matrix Riccati differential equation. The precise integration method based on the adaptive choosing of the two parameters is expanded from the traditional symmetric Riccati differential equation to the coupled asymmetric Riccati differential equation in this paper. The proposed expanded precise integration method can overcome the difficulty of the singularity point and the ill-conditioned matrix in the solving of coupled asymmetric Riccati differential equation. The numerical examples show that the expanded precise integration method gives more stable and accurate numerical results than the “direct integration method” and the “linear transformation method”.

scholarly journals Nonlinear Behavior of a Magnetic Bearing System

1995 ◽  
Vol 117 (3) ◽  
pp. 582-588 ◽  
Author(s):  
L. N. Virgin ◽  
T. F. Walsh ◽  
J. D. Knight
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Nonlinear Behavior ◽  
Analytical Techniques ◽  
Magnetic Bearing ◽  
Forced Response ◽  
The Mathematical Model ◽  
Two Degree Of Freedom ◽  
Sensitivity To Initial Conditions ◽  
Bearing System

This paper describes the results of a study into the dynamic behavior of a magnetic bearing system. The research focuses attention on the influence of nonlinearities on the forced response of a two-degree-of-freedom rotating mass suspended by magnetic bearings and subject to rotating unbalance and feedback control. Geometric coupling between the degrees of freedom leads to a pair of nonlinear ordinary differential equations, which are then solved using both numerical simulation and approximate analytical techniques. The system exhibits a variety of interesting and somewhat unexpected phenomena including various amplitude driven bifurcational events, sensitivity to initial conditions, and the complete loss of stability associated with the escape from the potential well in which the system can be thought to be oscillating. An approximate criterion to avoid this last possibility is developed based on concepts of limiting the response of the system. The present paper may be considered as an extension to an earlier study by the same authors, which described the practical context of the work, free vibration, control aspects, and derivation of the mathematical model.

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