A new dynamic model of rotor–blade systems (original) (raw)

The Influence of Shaft’s Bending on the Coupling Vibration of a Flexible Blade-Rotor System

Mathematical Problems in Engineering, 2017

The influence of shaft bending on the coupling vibration of rotor-blades system is nonignorable. Therefore, this paper analyzed the influence of shaft bending on the coupling vibration of rotor-blades system. The vibration mode function of shaft under elastic supporting condition was also derived to ensure accuracy of the model as well. The influence of the number of blades, the position of disk, and the support stiffness of shaft on critical speed of system was analyzed. The numerical results show that there were two categories of coupling mode shapes which belong to a set where the blade's first two modes predominate in the system: shaft-blade (SB) mode and interblade (BB) mode due to the coupling between blade and shaft. The BB mode was of repeated frequencies of (− 2) multiplicity for number blades, and the SB mode was of repeated frequencies of (2) multiplicity for number blades. What is more, with the increase of the number of blades, natural frequency of rotor was decreasing linearly, that of BB mode was constant, and that of SB mode was increasing linearly. Natural frequency of BB mode was not affected while that of rotor and SB mode was affected (changed symmetrically with the center of shaft) by the position of disk. In the end, vibration characteristics of coupling mode shapes were analyzed.

Linearly coupled shaft-torsional and blade-bending vibrations in multi-stage rotor-blade systems

Journal of Sound and Vibration, 2006

In an attempt towards the understanding of coupling effects between shaft-torsional and blade-(in plane) bending vibrations in turbomachinery and other rotating bladed structures, an idealized model is considered where the blades are represented by uniform Euler–Bernouilli beams. A synthetical, multi-frame and mixed (consisting in a coalescence of finite element and Galerkin methods) approach is used to derive a linearized mathematical model for the system. It is shown that the related eigenanalysis problem splits into two separate sub-problems corresponding to two kinds of possible normal mode motions. Eigenanalyses are performed concerning single as well as multi-stage rotor-blade system examples. Parametric studies are presented on the variation of the eigenfrequencies with system parameters. Eigenvalue loci veering phenomena are shown to occur, causing departure of the eigenvalues from those obtained through an uncoupled analysis. Mode shape examples are also presented.

GT2009-59471 ROTOR-BLADE COUPLED VIBRATION ANALYSIS BY MEASURING MODAL PARAMETERS OF ACTUAL ROTOR

The designers of rotor shafts and blades for a traditional turbinegenerator set typically employed their own models and process by neglecting the coupled torsional effect. The torsional coupled umbrella mode of recent longer blades systems designed for higher output and efficiency tends to have nearly doubled the frequency of electric disturbance (i.e., 100 or 120 Hz). In order to precisely estimate the rotor-blade coupled vibration of rotating shafts, the analysis must include a process to identify the parameters of a mathematical model by using a real model. In this paper we propose the use of a unique quasi-modal technique based on a concept similar to that of the modal synthesis method, but which represents a unique method to provide a visually reduced model. An equivalent mass-spring system is produced for uncoupled umbrella mode and modal parameters are measured in an actual turbine rotor system. These parameters are used to estimate the rotor-blade coupled torsional frequencies of a 700-MW turbine-generator set, with the accuracy of estimation being verified through field testing. NOMENCLATURE a coupling factor kij spring constant [K1], [K3] Stiffness matrix of shaft and blade k23 Coupling stiffness between shaft and blade [M1], [M3] Mass matrix of shaft and blade * m Modal mass of blade * c m Coupled modal mass of blade eq m , eq k Equivalent mass and equivalent spring constant p t I M m = = δ δ δ 3

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.

Nonlinear vibration response analysis of a rotor-blade system with blade-tip rubbing

An improved rotor-blade dynamic model is developed based on our previous works (Ma et al. in J Sound Vib, 337:301–320, 2015; J Sound Vib 357:168– 194, 2015). In the proposed model, the shaft is dis-cretized using a finite element method and the effects of the swing of the rigid disk and stagger angles of the blades are considered. Furthermore, the mode shapes of rotor-blade systems can be obtained based on the proposed model. The proposed model is more accurate than our previous model, and it is also verified by comparing the natural frequencies obtained from the proposed model with those from the finite element model and published literature. By simplifying the casing as a two degrees of freedom model, the single-and four-blade rubbings are studied using numerical simulation and experiment. Results show that for both the single-and four-blade rubbings, amplitude amplification phenomena can be observed when the multiple frequencies of the rotational frequency (f r) coincide with the conical and torsional natural frequencies of the rotor-blade system, natural frequencies of the casing and the bending natural frequencies of the blades. In addition, for the four-blade rubbing, the blade passing frequency (BPF, 4 f r) and its multiple frequency components also have larger amplitudes, especially, when they coincide with the natural frequencies of the rotor-blade system or casing; the four-blade rubbing levels are related to the rotor whirl, and the most severe rubbing happens on the blade located at the right end of the whirl orbit.

Mathematical model and simulator of rotor with vibrating blades

Journal of Konbin, 2014

The paper presents description of rotating bladed disk mathematical model. Correctly defined mathematical model of rotor allows creation of numerical simulation model which can be used to generate tip-timing data. First of all, the model is necessary to conduct a research on blade response due to input force in form of changing rotational speed. This enables the possibility to determine turbojet engine terminal operating conditions causing its failure.

Study on the Coupled Vibration Characteristics of a Two-Stage Bladed Disk Rotor System

Applied Sciences, 2021

This paper conducts a coupled vibration analysis of a two-stage bladed disk rotor system. According to the finite element method, the bladed disk rotor system is established. The substructure modal synthesis super-element method (SMSM) with a fixed interface and free interface is presented to obtain the vibration behaviors of the rotor system. Then, the free vibration results are compared with the ones calculated by the cyclic symmetry analysis method to validate the analysis in this paper. The results show that the modes of the two-stage bladed disk not only include the modes of the first- and second-stage bladed disk, but also the coupled modes of the two-stage bladed disk.

Contact analysis of a flexible bladed-rotor

European Journal of Mechanics A-solids, 2007

This paper presents a model of fully flexible bladed rotor developed in the rotating frame. An energetic method is used to obtain the matrix equations of the dynamic behaviour of the system. The gyroscopic effects as well as the spin softening effects and the centrifugal stiffening effects, taken into account through a pre-stressed potential, are included in the model. In the rotating frame, the eigenvalues' imaginary parts of the latter matrix equation give the Campbell diagram of the system and its stability can be analysed through its associated eigenvalues' real parts. The turbo machine casing is also modelled by an elastic ring in the rotating frame through an energetic method. Thus, in some rotational speed ranges the contact problem between the rotor and the stator can be treated as a static problem since both structures are stationary to each other. Prior to the study of the complete problem of contact between the flexible blades of the rotor and the flexible casing, a simple model of an elastic ring having only one mode shape, excited by rotating loads is developed in the rotating frame too, in order to underline divergence instabilities and mode couplings. Then, the complete problem of frictionless sliding contact between the blades and the casing, without rubbing, is studied. The stable balanced static contact configurations of the structure are found as function of the rotational speed of the rotor. Finally, the results are compared to these of the simple model of rotating spring-masses on an elastic ring, showing good adequacy. The present model of rotor appears thus particularly adapted to the study of blades-casing contacts and highlighted an unstable phenomenon near the stator critical speed even in case of frictionless sliding.

Model and Stability Analysis of a Flexible Bladed Rotor

International Journal of Rotating Machinery, 2006

This paper presents a fully bladed flexible rotor and outlines the associated stability analysis. From an energetic approach based on the complete energies and potentials for Euler-Bernoulli beams, a system of equations is derived, in the rotational frame, for the rotor. This later one is made of a hollow shaft modelled by an Euler-Bernoulli beam supported by a set of bearings. It is connected to a rigid disk having a rotational inertia. A full set of flexible blades is also modelled by Euler-Bernoulli beams clamped in the disk. The flexural vibrations of the blades as well as those of the shaft are considered. The evolution of the eigenvalues of this rotor, in the corotational frame, is studied. A stability detection method, bringing coalescence and loci separation phenomena to the fore, in case of an asymmetric rotor, is undertaken in order to determine a parametric domain where turbomachinery cannot encounter damage. Finally, extensive parametric studies including the length and the stagger angle of the blades as well as their flexibility are presented in order to obtain robust criteria for stable and unstable areas prediction.

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