CHAPTER 2 ANALYSIS OF SIMPLE ROTOR SYSTEMS (original) (raw)
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An experimental study evaluating parameters effects on the vibration of rotor
THE 1ST INTERNATIONAL CONFERENCE ON INNOVATIONS FOR COMPUTING, ENGINEERING AND MATERIALS, 2021: ICEM, 2021, 2021
Vibration in the rotary shaft is a common phenomenon observed in rotor systems. Determining the factors that cause the vibration to implement the vibration reduction measures to ensure the rotor is running stably is one of the urgent requirements today. In this paper, a machine model of rotor vibration assessment has been proposed, examining the main parameters influencing rotor vibrations, including critical speed, unbalance eccentricity and orientation angle. The experimental results show that: when the speed is around the critical speed, the vibration becomes unstable and the amplitude of the oscillation increases sharply; the orientation phase angle position causes different variations of the vibrating period leading to the imposed or superposition of tensile or compressive stresses resulting in fatigue occurring on the rotor. The present study illustrates the applicability of employing simple models to predict the dynamic response of a simple rotor system with acceptable accuracy.
7th IFToMM-Conference on Rotor Dynamics, Vienna, Austria, 25-28
2006
In the paper the structural one-dimensional hybrid dynamical model of the entire vibrating rotor-shaft system and the three-dimensional finite element model of its cracked shaft zone were applied for a fatigue life prediction of the machine faulty segment under coupled bending-torsional-axial vibrations. The steady-state dynamic response amplitudes, obtained by means of the one-dimensional model of the system, have been used for the three-dimensional model as an input data for determination of maximal stresses and stress intensity factors at the crack tip. These quantities together with the Wöhler curves enable us an approximate determination of load limits responsible for a probable further crack propagation. By means of the proposed approach one can predict a damage probability of the faulty rotor-shaft system of arbitrary structure operating under various dynamic and quasi-static loads affecting a crack of various sizes and shaft locations. From the investigations performed for v...
A Practical Guide to Rotor Dynamics
2004
Introduction Rotor dynamics is a very interesting and complicated subject. The importance of this subject has increased over the last few decades as machine speeds have increased and higher flows and efficiencies have had the side effect of introducing problems with critical speeds, unbalance response and rotor stability. Some of these problems are related to the economics of the capital expenditures where a machine may be bought on a cost basis. Far too often this results in excessive and expensive rebuilds and costly lost production. This paper will introduce the basics of rotor dynamics and lead into some of the more advanced concepts. The mathematics will be kept to a minimum and as many helpful "rules of thumb" will be included as this subject allows. Units are shown in detail for each equation. At times, rotor dynamics can be a very controversial subject from the nomenclature used to the question of the degree of accuracy needed to model a rotor dynamic system. There...
Rotordynamic Analysis in the Design of Rotating Machinery
2000
Rotordynamic analysis is an important step in the design of any rotating machine. To go beyond the very simple models yielding a good qualitative insight but cannot predict the details of the dynamic behaviour of rotors, it is necessary to resort to numerical methods and among them the Finite Element Method is without doubt the most suited for implementation in the context of computer aided engineering. Instead of resorting to general purpose codes, the particular characteristics of rotordynamic analysis make it expedient to use specialised tools like DYNROT, a FEM code which allows to perform a complete study of the dynamic behaviour of rotors. Although initially designed to solve the basic linear rotordynamic problems (Campbell diagram for damped or undamped systems, unbalance response, critical speeds, static loading), it has been extended to the study of nonstationary motions of nonlinear rotating systems [1] and the torsional and axial analysis of rotors and reciprocating machines. Its distinctive features of resorting to Guyan reduction and of extensively using complex coordinates both for isotropic and non symmetric systems, allow to reduce the computer time and to perform a large number of computations at a reasonable cost. The code can thus be used as a routine to be called by optimisation procedures aimed at including rotordynamic performances into the definition of an optimum design of the machine.
Modeling of Dynamic Rotors with
2016
Modeling of Dynamic Rotors with Flexible Bearings due to the use of Viscoelastic Materials Nowadays rotating machines produce or absorb large amounts of power in relatively small physical packages. The fact that those machines work with large density of energy and flows is associated to the high speeds of rotation of the axis, implying high inertia loads, shaft deformations, vibrations and dynamic instabilities. Viscoelastic materials are broadly employed in vibration and noise control of dynamic rotors to increase the area of stability, due to their high capacity of vibratory energy dissipation. A widespread model, used to describe the real dynamic behavior of this class of materials, is the fractional derivative model. Resorting to the finite element method it is possible to carry out the modeling of dynamic rotors with flexible bearings due to the use of viscoelastic materials. In general, the stiffness matrix is comprised of the stiffnesses of the shaft and bearings. As considered herein, this matrix is complex and frequency dependent because of the characteristics of the viscoelastic material contained in the bearings. Despite of that, a clear and simple numerical methodology is offered to calculate the modal parameters of a simple rotor mounted on viscoelastic bearings. A procedure for generating the Campbell diagram (natural frequency versus rotation frequency) is presented. It requires the embedded use of an auxiliary (internal) Campbell diagram (natural frequency versus variable frequency), in which the stiffness matrix as a frequency function is dealt with. A simplified version of that procedure, applicable to unbalance excitations, is also presented. A numerical example, for two different bearing models, is produced and discussed
Dynamics of a Simple Rotor System
This paper describes how to develop a mathematical model of a rotordynamical system for a reader either is a beginner or has a background in dynamical systems. The theory is described in order to conceive the main idea how to develop dynamical models of rotors. The problem is based on FEM and described in detail. The comparison of results is done with help of MATLAB code using the ROTORDYNAMIC TOOLBOX and Ansys modal analysis. The purpose of this paper, as mentioned above, is to identify why calculations are not the same or sometimes not even close at all to the real values. Thus, the main goal is, to track down what more should be taken into consideration in the computation (like bearings, support and so forth).
Study of the lateral vibration in a multistage rotor
Revista Facultad de Ingeniería Universidad de Antioquia, 2020
This paper presents the development of theoretical and experimental models for the study of rotodynamic behavior of a multistage rotor. The transfer matrix method, which includes the characteristics of stiffness and damping for the supports and the stages respectively as well as the characteristics of unbalance in the stages, is used for the theoretical model. The data from a physical model was employed as a way of validating the theoretical results. The first two critical speeds were determined with the theoretical model and they differ in a low percentage with respect to the values measured experimentally. Moreover, the vibration level recorded in the physical model rises 2.5 times when the multistage rotor approaches the first two critical speeds. In addition to this, significant displacements of the lateral critical speeds are noticeable when an increase in mass imbalance is induced in several of the rotor impellers. RESUMEN: Este artículo presenta el desarrollo de modelos, teórico y experimental, para el estudio del comportamiento rotodinámico de un rotor multietapas. El modelo teórico se obtiene a partir de las ecuaciones de Lagrange y se resuelve empleando el método de la matriz de transferencia. En el análisis se incluyen las características de rigidez y amortiguamiento de los apoyos y de las diferentes etapas del rotor, así como el desbalance másico de estas últimas. Como vía de validación de los resultados teóricos se emplearon los datos provenientes de un modelo físico rotodinámico, especialmente desarrollado para la investigación. Las dos primeras velocidades críticas fueron determinadas con el modelo teórico y difieren en un bajo porcentaje con respecto a los valores medidos experimentalmente. Por otro lado, el nivel de vibración registrado en el modelo físico se eleva 2,5 veces cuando el rotor multietapas se aproxima a cualquiera de las primeras dos velocidades críticas. Adicionalmente, se describe un corrimiento apreciable en la magnitud de las velocidades críticas laterales cuando se aumenta el desbalance másico en varios de los impulsores del rotor.
A new dynamic model of rotor–blade systems
A new dynamic model of rotor-blade systems is developed in this paper considering the lateral and torsional deformations of the shaft, gyroscopic effects of the rotor which consists of shaft and disk, and the centrifugal stiffening, spin softening and Coriolis force of the blades. In this model, the rotating flexible blades are represented by Timoshenko beams. The shaft and rigid disk are described by multiple lumped mass points (LMPs), and these points are connected by massless springs which have both lateral and torsional stiffness. LMPs are represented by the corresponding masses and mass moments of inertia in lateral and rotational directions, where each point has five degrees of freedom (dofs) excluding axial dof. Equations of motion of the rotor-blade system are derived using Hamilton's principle in conjunction with the assumed modes method to describe blade deformation. The proposed model is compared with both finite element (FE) model and real experiments. The proposed model is first validated by comparing the model natural frequencies and vibration responses with those obtained from an FE model. A further verification of the model is then performed by comparing the model natural frequencies at zero rotational speed with those obtained from experimental studies. The results shown a good agreement between the model predicted system characteristics and those obtained from the FE model and experimental tests. Moreover, the following interesting phenomena have been revealed from the new model based analysis: The torsional natural frequency of the system decreases with the increase of rotational speed, and the frequency veering phenomenon has been observed at high rotational speed; The complicated coupling modes, such as the blade-blade coupling mode (BB), the coupling mode between the rotor lateral vibration and blade bending (RBL), and the coupling mode between the rotor torsional vibration and blade bending (RBT), have also been observed when the number of blades increases.
In this paper, a continuous rotor system is modelled by considering some critical factors, like the gyroscopic and rotary inertia effects of disc and shaft cross-sections, large shaft deformation, and restriction to shaft axial motion at the bearings. The bearings are replaced by springs along horizontal and vertical directions. Governing partial differential equations (PDE’s) for the vibrations of the disc along the horizontal and vertical directions are derived by employing the Hamiltonian principle. The governing equation is then transformed into a set of ordinary differential equations (ODEs) using method of modal projection. The large deformation and restriction to axial motion of the shaft yields nonlinearities in the system governing equations. Only the linear system is analysed in this first part. The parameters in the dimensionless form of the governing equations are functions of some independent variables which are associated to the material and geometrical properties of t...