An Application of Torsional Wave Analysis to Turbogenerator Rotor Shaft Response

1992 ◽  
Vol 114 (2) ◽  
pp. 149-153 ◽  
Author(s):  
R. Bogacz ◽  
T. Szolc ◽  
H. Irretier
Keyword(s):  
Cross Sections ◽  
High Speed ◽  
Equations Of Motion ◽  
Wave Theory ◽  
Continuous Model ◽  
Rotor Shaft ◽  
Shaft System ◽  
Stepped Shaft ◽  
Angular Velocities ◽  
Turbogenerator Rotor

In this paper transient torsional vibrations of a steam turbogenerator rotor shaft system due to high speed reclosing of the electric network are investigated. The analysis is performed using torsional elastic wave theory applied to a continuous model in the form of a stepped shaft. Wave solutions of the equations of motion are used in order to determine dynamic torsional elastic moments and vibratory angular velocities in cross-sections of the turbine shafts. The results are illustrated in the form of graphs.

Stability analysis of rotor whirl under nonlinear internal friction by a general averaging approach

2016 ◽  
Vol 23 (5) ◽  
pp. 808-826 ◽  
Author(s):  
Francesco Sorge
Keyword(s):  
Internal Friction ◽  
Equations Of Motion ◽  
Translational Motion ◽  
Stable Limit ◽  
Rotor Shaft ◽  
Shaft System ◽  
Flexible Supports ◽  
Asymmetric Rotor ◽  
Nonlinear Equations Of Motion ◽  
Shrink Fit

The two main sources of internal friction in a rotor-shaft system are the shaft structural hysteresis and the possible shrink-fit release of the assembly. The internal friction tends to destabilize the over-critical rotor running, but a remedy against this effect may be provided by a proper combination of some external damping in the supports and an anisotropic arrangement of the support stiffness, or at most by the support damping alone, depending on the system geometry. The present analysis reported here considers a general asymmetric rotor-shaft system, where the rotor is perfectly rigid and is constrained by viscous–flexible supports having different stiffnesses on two orthogonal planes. The internal friction is modelled by nonlinear Coulombian forces, which counteract the translational motion of the rotor relative to a frame rotating with the shaft ends. The nonlinear equations of motion are dealt with using an averaging approach based on the Krylov-Bogoliubov method with some adaptation to address the multi-degree-of-freedom nature of the problem. Stable limit cycles may be attained by the overcritical whirling motions, whose amplitudes are inversely proportional to the external dissipation applied by the supports. A noteworthy result is that the stiffness anisotropy of the supports is recognized as beneficial in reducing the natural whirl amplitudes, albeit mainly in the symmetric configuration of the rotor at the mid span and, to a rather lesser extent, in the asymmetric configuration, which then requires a stronger damping action in the supports.

Parametric resonances for torsional vibration of excited rotating machineries with nonconstant velocity joints

2017 ◽  
Vol 24 (15) ◽  
pp. 3262-3277 ◽  
Author(s):  
Masoud SoltanRezaee ◽  
Mohammad-Reza Ghazavi ◽  
Asghar Najafi
Keyword(s):  
Torsional Vibration ◽  
High Speed ◽  
Power Transmission ◽  
Equations Of Motion ◽  
Monodromy Matrix ◽  
Parametric Instability ◽  
Constant Input ◽  
Inertia Moment ◽  
Disk Model ◽  
Shaft System

The shaft system is a rotating machinery with many applications due to its high speed. The angle between shafts may not be zero. So the shafts can be connected to each other through a nonconstant velocity U-joint, which transforms a constant input angular velocity into a periodically fluctuating velocity. Consequently, the mechanism is parametrically excited and may face resonance conditions. Herein, a power transmission system including three elastic shafts is considered. The polar inertia moment of each shaft is modeled as a dynamic system with two discrete disks at the shaft ends. The equations of motion consist of a set of Mathieu–Hill differential equations with periodic coefficients. The dynamic stability and torsional vibration of the shaft system are analyzed. The system geometry and inertia moment effect are the main issues in this contribution. Parametric instability charts are achieved via the monodromy matrix technique. The graphical numerical results are validated with the frequency analytical results. Finally, the stability regions are shown in the parameter spaces of velocity, misalignment angles and the inertia of disks. The results demonstrated that by changing the system inertia and geometry, stabilizing the whole system is possible. Moreover, to check the precision of the model, the results are compared with a basic single-disk model, which is prevalent in two-shaft systems.

Viscoelastic modelling of rotor—shaft systems using an operator-based approach

2010 ◽  
Vol 225 (1) ◽  
pp. 73-87 ◽  
Author(s):  
J K Dutt ◽  
H Roy
Keyword(s):  
Equations Of Motion ◽  
Broad Band ◽  
Viscoelastic Model ◽  
Stability Limit ◽  
Material Model ◽  
Time Response ◽  
Constitutive Relationship ◽  
Rotor Shaft ◽  
Forcing Function ◽  
Shaft System

Damping exists in every material in varying degrees, so materials in general are viscoelastic in nature. Energy storage, as well as dissipation in varying degrees, accompanies every time-varying deformation, with the effect that stress and strain in a material get out of phase. This work presents the development of equations of motion of a rotor—shaft system with a viscoelastic rotor after discretizing the system into finite elements. Subsequently, these equations are used to study the dynamics of the rotor—shaft system in terms of stability limit of spin speed and time response of a disc as a result of unbalance. The primary inspiration for a viscoelastic model arises from the need to capture the influence of broad band spectral behaviour of rotor—shaft materials, primarily polymers and polymer composites, which are principally the materials of light rotors, on the dynamics of rotor—shaft system. For this, the material constitutive relationship has been represented by a differential time operator. Use of operators enables one to consider general linear viscoelastic behaviours, represented in the time domain by multi-element (three, four, or higher elements) spring—dashpot models or internal variable models, for which, in general, instantaneous stress and its derivatives are proportional to instantaneous strain and its derivatives. Again such representation is fairly generic, in a sense that the operator may be suitably chosen according to the material model to obtain the equations of motion of a rotor—shaft system. The equations so developed may be easily used to find the stability limit speed of a rotor—shaft system as well as the time response when the rotor—shaft system is subjected to any dynamic forcing function.

Basic Study on the Seismic Response of the High-Speed Rotational Body Supported by Fluid Film Bearings

2011 ◽  
Author(s):  
Masayoshi Hatta ◽  
Atsuhiko Shintani ◽  
Tomohiro Ito
Keyword(s):  
High Speed ◽  
Seismic Waves ◽  
Equations Of Motion ◽  
Rotating Disk ◽  
Fluid Film ◽  
Basic Study ◽  
Fluid Forces ◽  
Shaft System ◽  
Fluid Film Bearings ◽  
Response Behaviors

In this study, the seismic responses of a disk and a shaft are evaluated analytically. In an analytical model, the disk-shaft system is treated as an elastic shaft with a rigid disk, and the shaft is supported by fluid film bearings. Furthermore, the gyroscopic effect of a disk and the fluid forces due to fluid film bearings are considered. The equations of motion are derived for the translational and rotational motions when the floor is subjected to horizontal and vertical excitations. The displacements of the centers of the disk and the shaft are evaluated by numerical simulations. At first, the response behaviors of a rotating disk without base excitation are evaluated, and at second, the effects of sinusoidal base excitations are investigated. Finally, the response behaviors of this system are subjected to seismic waves of varying frequencies. The results of the different seismic wave input are studied.

Optimal Parameters of an Electromagnetic Actuator for Actively Controlling the Vibration of a Flexible Rotor-Shaft System

2010 ◽  
Author(s):  
Anindya S. Das ◽  
Tarapada Roy ◽  
Jayanta K. Dutt
Keyword(s):  
Finite Element ◽  
Equations Of Motion ◽  
Electromagnetic Actuator ◽  
Flexible Rotor ◽  
Control Cost ◽  
Rotor Shaft ◽  
Shaft System ◽  
And Control ◽  
Optimal Set ◽  
Bearing System

The present work deals with finding the optimum parameter of a control system that utilizes an electromagnetic actuator to actively attenuate the vibration amplitude of a flexible rotor-shaft-bearing system. The equations of motion of the rotor-shaft-bearing system is found out using the finite element method and the rotor-shaft is modelled by beam finite element taking into account of the effects like distributed inertia, flexural stiffness, gyroscopic effect and internal material damping. Optimum actuator location and the optimal set of actuator and control parameters are found out using Genetic Algorithm based approach, where the objective is to minimize response amplitude with least amount of control cost with a specified margin of stability. The control parameter found in this approach is quite efficient in achieving the desired objective.

Study on the Seismic Response of High-Speed Rotational Disk-Shaft System

2012 ◽  
Author(s):  
Tomohiro Ito ◽  
Masayoshi Hatta ◽  
Atsuhiko Shintani ◽  
Chihiro Nakagawa
Keyword(s):  
Seismic Response ◽  
Seismic Wave ◽  
High Speed ◽  
Equations Of Motion ◽  
Rotating Disk ◽  
Seismic Responses ◽  
Shaft System ◽  
High Speed Rotation ◽  
Gyroscopic Effects ◽  
Response Behaviors

Recently, huge earthquakes occurred in the world, such as Great East Japan Earthquake in Japan or huge earthquake in New Zealand in 2011. In Niigata-ken Chuetsuoki earthquake in 2007, turbine blade collisions with the casing were found. Therefore, it is very important to clarify the seismic response behaviors of the high-speed rotational shaft like turbine shafts, because high-speed rotation will cause characteristic dynamic effects, such as gyro moment. In this study, seismic responses of a disk-shaft system are evaluated analytically. The disk-shaft system is treated as an elastic shaft with a rigid disk. In this analysis, gyroscopic effects of a rotating disk are also considered. Equations of motion are derived for the translational and rotational motions when the system is subjected to both horizontal and vertical seismic excitations. Response behaviors of a rotating disk-shaft system are evaluated for sinusoidal excitations and seismic wave excitations. In the sinusoidal excitations, excitation frequencies and rotational speed are varied, and in the seismic wave excitations, seismic responses are evaluated for the waves with various predominant frequencies.

Analysis of Coupled Vibration Response in a Rotating Flexible Shaft-Impeller System

1980 ◽  
Vol 102 (1) ◽  
pp. 162-167 ◽  
Author(s):  
N. Hagiwara ◽  
S. Sakata ◽  
M. Takayanagi ◽  
K. Kikuchi ◽  
I. Gyobu
Keyword(s):  
High Speed ◽  
Vibration Response ◽  
Simple Method ◽  
Nodal Diameter ◽  
Rotor Shaft ◽  
Dynamical Characteristics ◽  
Shaft System ◽  
Transfer Method ◽  
Dynamical Connection ◽  
Theoretical Results

This paper presents a way of analyzing the vibration of a rotor shaft system coupled with flexible impellers based on the transfer method. Each of the flexible impellers is modelled so that it comprises an inertia and an elastic hinge based on the assumption that any impeller’s vibration mode except its one-nodal diameter has no dynamical connection with a shaft. The stiffness of the hinge is based on the eigen frequency of its mode calculated by means of a finite element method, or obtained by experiment. The method developed allows the calculation of the vibration response of a rotor shaft system consisting of several flexible impellers and several bearings when it is subject to an out-of-balance force. Theoretical results are compared with experimental data and show very satisfactory agreement. This simple method appears therefore to be practical and efficient to predict the dynamical characteristics of a high speed rotor system with coupling of impeller flexibility.

A Second Order Lagrangian Model for Irregular Ocean Waves

2005 ◽  
Vol 128 (3) ◽  
pp. 177-183 ◽  
Author(s):  
Sébastien Fouques ◽  
Harald E. Krogstad ◽  
Dag Myrhaug
Keyword(s):  
Specular Reflection ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Wave Theory ◽  
Ocean Waves ◽  
Second Order ◽  
Lagrangian Model ◽  
Lagrangian Description ◽  
Linear Wave ◽  
Irregular Solution

Synthetic aperture radar (SAR) imaging of ocean waves involves both the geometry and the kinematics of the sea surface. However, the traditional linear wave theory fails to describe steep waves, which are likely to bring about specular reflection of the radar beam, and it may overestimate the surface fluid velocity that causes the so-called velocity bunching effect. Recently, the interest for a Lagrangian description of ocean gravity waves has increased. Such an approach considers the motion of individual labeled fluid particles and the free surface elevation is derived from the surface particles positions. The first order regular solution to the Lagrangian equations of motion for an inviscid and incompressible fluid is the so-called Gerstner wave. It shows realistic features such as sharper crests and broader troughs as the wave steepness increases. This paper proposes a second order irregular solution to these equations. The general features of the first and second order waves are described, and some statistical properties of various surface parameters such as the orbital velocity, slope, and mean curvature are studied.

Transient Vibration of High-Speed, Lightweight Rotors Due to Sudden Imbalance

1983 ◽  
Vol 105 (3) ◽  
pp. 480-486 ◽  
Author(s):  
M. Sakata ◽  
T. Aiba ◽  
H. Ohnabe
Keyword(s):  
Transient Response ◽  
High Speed ◽  
Rotor System ◽  
Response Analysis ◽  
Transient Vibration ◽  
Flexible Shaft ◽  
Shaft System ◽  
Transient Response Analysis ◽  
Flexible Disk ◽  
Blade Loss

In the field of rotor dynamics, increased attention is being given to the transient response analysis of the rotor, since the effects of impact loading and vibrations of the rotor arising from blade loss can be studied by a time transient solution of the rotor system. As recent trends in rotating machinery have been directed towards lightweight, high-speed flexible rotors, the effect of flexibility on transient response analysis is becoming of increasing importance. In the present paper, a transient vibration analysis is carried out on a flexible-disk/flexible-shaft system or rigid-disk flexible-shaft system subjected to a sudden imbalance that is assumed to represent the effect of blade loss. To solve the basic equation governing a rotating flexible disk the Galerkin’s method is used, and the equation of motion of the rotor system is numerically solved by employing the Runge-Kutta-Gill’s method. Experiments were conducted on a model rotor having a blade loss simulator; the shaft vibrations were also measured. The validity of the anaytical results was demonstrated by comparison with the experimental results.

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