TUNING OF SIMULATED NATURAL FREQUENCIES FOR A FLEXIBLE SHAFT–MULTIPLE FLEXIBLE DISK SYSTEM

AbstractThis study investigates the effects of disk position nonlinearities on the nonlinear dynamic behavior of a rotating flexible shaft-disk system. Displacement of the disk on the shaft causes certain nonlinear terms which appears in the equations of motion, which can in turn affect the dynamic behavior of the system. The system is modeled as a continuous shaft with a rigid disk in different locations. Also, the disk gyroscopic moment is considered. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed modes method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work are inclusive of time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The effect of disk nonlinearities is studied for some disk positions. The results confirm that when the disk is located at mid-span of the shaft, only the regular motion (period one) is observed. However, periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for situations in which the disk is located at places other than the middle of the shaft. The results show nonlinear effects are negligible in some cases.

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

Journal of Engineering for Power ◽

10.1115/1.3227440 ◽

1983 ◽

Vol 105 (3) ◽

pp. 480-486 ◽

Cited By ~ 11

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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Additional Investigations Into the Natural Frequencies and Critical Speeds of a Rotating, Flexible Shaft-Disk System

Volume 5: Turbo Expo 2007 ◽

10.1115/gt2007-28065 ◽

2007 ◽

Cited By ~ 3

Author(s):

Lyn M. Greenhill ◽

Valerie J. Lease

Keyword(s):

Natural Frequencies ◽

Slenderness Ratio ◽

Rotor Dynamics ◽

Axial Position ◽

Disk System ◽

Apparent Contradiction ◽

Critical Speeds ◽

Shaft Flexibility ◽

Gyroscopic Effects ◽

Lumped Masses

Traditional rotor dynamics analysis programs make the assumption that disk components are rigid and can be treated as lumped masses. Several researchers have studied this assumption with specific analytical treatments designed to simulate disk flexibility. The general conclusions reached by these studies indicated disk flexibility has little effect on critical speeds but significantly influences natural frequencies. This apparent contradiction has been reexamined by using axisymmetric harmonic finite elements to directly represent both disk and shaft flexibility along with gyroscopic effects. Results from this improved analysis show that depending on the thickness-to-diameter (slenderness) ratio of the disk and the axial position of the disk on the shaft, there are significant differences in all natural frequencies, for both forward and backward modes, including synchronous crossings at critical speeds.

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Forced Harmonic Response of Grouped Blade Systems: Part II—Application

Journal of Engineering for Gas Turbines and Power ◽

10.1115/1.2816529 ◽

1996 ◽

Vol 118 (1) ◽

pp. 137-145 ◽

Cited By ~ 8

Author(s):

L. F. Wagner ◽

J. H. Griffin

Keyword(s):

Numerical Model ◽

Vibration Analysis ◽

Natural Frequencies ◽

Turbine Blades ◽

Modal Identification ◽

Harmonic Response ◽

Model Based ◽

Grouped Blades ◽

Flexible Disk ◽

Symmetric Structures

The vibration of grouped blades on a flexible disk should, for purposes of economy and clarity of modal identification, be analyzed using procedures developed for cyclically symmetric structures. In this paper, a numerical model, based on the theory of cyclically symmetric structures, is applied to the vibration analysis, and in particular, the harmonic response, of a flexible disk supporting a number of groups, or packets, of turbine blades. Results are presented to show variations in the modal participation factors as a function of such parameters as disk flexibility, blade density, and the total number of assembled groups. It is also shown that many characteristics of the system spectra of natural frequencies are strongly dependent on the number of blade groups.

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The Influences of Stagger and Pretwist Angles of Blades on Coupling Vibration in Shaft-Disk-Blade Systems

Journal of Vibration and Acoustics ◽

10.1115/1.4044991 ◽

2019 ◽

Vol 142 (1) ◽

Author(s):

Hassan Heydari ◽

Amir Khorram ◽

Laya Afzalipour

Keyword(s):

Aspect Ratio ◽

Natural Frequencies ◽

Thickness Ratio ◽

Lagrangian Approach ◽

Coupling Vibration ◽

Flexible Shaft ◽

Blade Thickness ◽

Out Of Plane ◽

Coupling Pattern ◽

Stagger Angle

Abstract The influences of stagger angle (α) and pretwist angle (βL) of blades on the coupling vibration among shaft bending and blade bending in a shaft-disk-blade (SDB) system are investigated using a Lagrangian approach in combination with the assumed modes method (AMM). The disk is rigid, and the flexible shaft is supported with two rigid bearings. It is shown that α and βL have variable effects on the coupling vibration because their influences can be increased, reduced, or even completely eliminated for different values of disk location (λ), blade thickness ratio (δ), and blade aspect ratio (γ). To study the coupling vibration in an SDB system, consideration of λ, δ, and γ are very important because those can alter the coupling magnitude, the coupling pattern as well as the predominant modes. Nevertheless, previous researches rarely take into account these parameters. Moreover, in the present work, to investigate the natural frequencies and critical speeds versus λ, δ, and γ, new diagrams are introduced. Also, the relation between the in-plane and out-of-plane motions of the blades with the coupling vibration is precisely analyzed.

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Dynamic Formulations and Energy Analysis of Rotating Flexible-Shaft/Multi-Flexible-Disk System With Eddy-Current Brake

Journal of Vibration and Acoustics ◽

10.1115/1.1309538 ◽

2000 ◽

Vol 122 (4) ◽

pp. 365-375 ◽

Cited By ~ 2

Author(s):

Rong-Fong Fung ◽

Shih-Ming Hsu

Keyword(s):

Eddy Current ◽

Energy Analysis ◽

Virtual Work ◽

Beam Theory ◽

Brake System ◽

Euler Beam ◽

Disk System ◽

Coupling System ◽

Flexible Disk ◽

Work Done

In this paper, the rotating flexible-Timoshenko-shaft/flexible-disk coupling system is formulated by introducing the kinetic and strain energies, and the virtual work done by the eddy-current brake system into Hamilton’s principle. The attachment of disk to shaft becomes flexible for Timoshenko-beam theory and rigid for Euler-beam theory. It is found that the eddy-current brake system can be used to decrease speed and suppress flexible and shear vibrations simultaneously. From the dynamic formulations and energy analysis, some important discussions are made. Numerical results are provided to validate the theoretical analysis. [S0739-3717(00)01504-X]

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