Stability of a Cracked Rotor Subjected to Parametric Excitation

2015 ◽  
Vol 137 (5) ◽  
Author(s):  
Jerzy T. Sawicki ◽  
Zbigniew Kulesza
Keyword(s):  
Parametric Excitation ◽  
Crack Detection ◽  
Averaging Method ◽  
Cracked Rotor ◽  
Parametric Vibrations ◽  
Excited System ◽  
Artificial Damping ◽  
Parametric Resonances ◽  
The Stability ◽  
Cracked Shaft

It is well known that parametric vibrations may appear during the rotation of a rotor with a cracked shaft. The vibrations occur due to periodic stiffness changes being the result of the crack breathing. A parametrically excited system may exhibit parametric resonances and antiresonances affecting the stability of the system. In most cases the destabilizing effect due to parametric resonances is studied. Antiresonant cases seem to be uninteresting. However, the antiresonances have a unique property of introducing additional artificial damping to the system, thus improving its stability and reducing the vibration amplitude. Apart from different control applications, this stabilizing effect may be interesting for its probable ability to indicate the shaft crack. The possible application of the additional damping introduced by parametric excitation for the shaft crack detection is analyzed in the present paper. The approach is demonstrated with a mathematical model of a rotor with a cracked shaft. The stability analysis of the rotor is performed analytically by employing the averaging method. Stability boundaries for different frequencies of the parametric excitation and for different crack depths are derived. The results of this analysis are checked numerically by means of the Floquet's theory. Next, possible applications of the parametric excitation for the shaft crack detection are validated numerically.

Stability of a Cracked Rotor Subjected to Parametric Excitation

2014 ◽  
Author(s):  
Jerzy Sawicki ◽  
Zbigniew Kulesza
Keyword(s):  
Parametric Excitation ◽  
Crack Detection ◽  
Experimental Studies ◽  
Interesting Property ◽  
Suggested Approach ◽  
Excited System ◽  
Artificial Damping ◽  
Parametric Resonances ◽  
The Stability ◽  
Cracked Shaft

It is well known that parametric vibrations may appear during the rotation of a rotor with a cracked shaft. The vibrations occur due to periodic stiffness changes being the result of crack breathing. A parametrically excited system may exhibit parametric resonances and anti-resonances affecting the stability of the system. In most cases the destabilizing effect due to parametric resonances is studied. Anti-resonant cases seem to be uninteresting. However, the anti-resonances have an interesting property of introducing additional artificial damping to the system, thus improving its stability and reducing the vibration amplitude. Apart from different control applications, this stabilizing effect may be interesting for its probable ability to indicate the shaft crack. The possible application of the additional damping introduced by parametric excitation for the shaft crack detection is analyzed in the present paper. The approach is demonstrated with a mathematical model of a rotor with a cracked shaft. The stability analysis of the rotor is performed analytically by employing the averaging method. Stability boundaries for different frequencies of the parametric excitation and for different crack depths are derived. The results of this analysis are checked numerically by means of the Floquet’s theory. Next, possible applications of the parametric excitation for the shaft crack detection are validated numerically. The results demonstrate the potential of the suggested approach, yet further analytical and experimental studies are required.

Parametric Vibrations in a Magneto-Thermo-Elastic Layer of Finite Thickness

1995 ◽  
Author(s):  
Jörg Wauer
Keyword(s):  
Magnetic Field ◽  
Finite Thickness ◽  
Elastic Field ◽  
Steady State Response ◽  
Bias Magnetic Field ◽  
Parametric Vibrations ◽  
State Response ◽  
Parametric Resonances ◽  
The Stability ◽  
The Magnetic Field

Abstract The dynamic behavior of a magneto-thermo-elastic planar layer subjected to a bias magnetic field is examined. The magnetic field acts parallel to the layer surfaces and is composed of a constant and a harmonically oscillating part. In particular, the small coupled magneto-thermo-elastic vibrations superimposed on the basic steady state response due to the stationary magnetic excitation are analyzed. Attention is focused on the influence of the pulsating part of the bias magnetic field on the governing variational equations to prove the stability of the basic state. In general, the stability equations describing the perturbations of the magnetic, thermal and elastic field properties are coupled in a special form and also the parametric excitation acts in a non-classical manner. Hence the variety of possible parametric resonances seems to be limited.

The Dynamics of a Rotor System with a Cracked Shaft

1986 ◽  
Vol 108 (2) ◽  
pp. 189-196 ◽  
Author(s):  
H. D. Nelson ◽  
C. Nataraj
Keyword(s):  
Computer Code ◽  
Fourier Series Expansion ◽  
Simulation Studies ◽  
Cracked Rotor ◽  
Rotating Shaft ◽  
Crack Location ◽  
Excited System ◽  
System Equations ◽  
Stiffness Variation ◽  
Cracked Shaft

A theoretical analysis of the dynamics of a rotor-bearing system with a transversely cracked rotor is presented. The rotating assembly is modeled using finite rotating shaft elements and the presence of a crack is taken into account by a rotating stiffness variation. This stiffness variation is a function of the rotor’s bending curvature at the crack location and is represented by a Fourier series expansion. The resulting parametrically excited system is nonlinear and is analyzed using a perturbation method coupled with an iteration procedure. The system equations are written in terms of complex variables and an associated computer code has been developed for simulation studies. Results obtained by this analysis procedure are compared with previous analytical and experimental work presented by Grabowski.

Coupled Lateral and Torsional Vibrations of a Cracked Rotor

2004 ◽  
Author(s):  
Jerzy T. Sawicki ◽  
George Y. Baaklini ◽  
Andrew L. Gyekenyesi
Keyword(s):  
Crack Detection ◽  
Vibration Spectrum ◽  
Equations Of Motion ◽  
Structural Response ◽  
Crack Model ◽  
Cracked Rotor ◽  
Lateral Vibrations ◽  
Crack Problems ◽  
Torsional Excitation ◽  
Cracked Shaft

Rotor crack problems present a significant safety and loss hazard in nearly every application of modern turbomachinery, particularly in the power generation industry. However, early crack detection is not easily achieved during the operation of machinery. The difficulty is based on the fact that a crack produces an undetectable change in the overall structural response. This paper analyzes the coupling of torsional and lateral vibrations for an unbalanced cracked rotor. The rotor equations of motion for a system with cracked shaft, obtained using Lagrangian dynamics, show coupling and nonlinear interaction between the torsional and lateral vibrations. To investigate the effect of a transverse surface crack on the dynamic rotor response the breathing crack model was employed. By applying an external torsional excitation together with the excitation due to unbalance, signature responses were observed in the rotor vibration spectrum at sum and difference frequencies. These signature responses were due to the nonlinear effect of the crack. The observed phenomena, analytically defined here, offers a new methodology concerning crack detection and prognosis in rotors.

scholarly journals Stability Analysis of a Breathing Cracked Rotor with Imposed Mass Eccentricity

2015 ◽  
Vol 2015 ◽  
pp. 1-17 ◽  
Author(s):  
Huichun Peng ◽  
Qing He ◽  
Yaxin Zhen
Keyword(s):  
Crack Detection ◽  
Rotation Speed ◽  
Rotor System ◽  
Critical Ratio ◽  
Cracked Rotor ◽  
Stability Diagrams ◽  
Mass Eccentricity ◽  
Stability And Bifurcations ◽  
The Stability ◽  
Cracked Rotor System

The investigation of the effects of mass eccentricity on stability of a gravity dominated cracked Jeffcott rotor would generally provide practical applicability to crack detection and instability control of the heavy loading turbo-machinery system. Based upon the numerical Floquet method, the stability and bifurcations of the periodic time-dependent rotor with a transverse breathing crack are studied with respect to the varied mass eccentricity at different rotation speed, and the stability diagrams in the parameter plane are obtained which the previous studies have not covered. The numerical response of the cracked rotor system is also analyzed by the frequency spectrum to present the vibration characters while the rotation speed approaches the critical ratio. The detailed numerical eigenvalues of the transition matrix are applied to analyze the types of the bifurcations of the cracked rotor system. Three types of bifurcations are found and responses of the cracked rotor system at these bifurcations are presented for the visualized comparisons.

scholarly journals Vibration and Stability Analysis of a Bearing–Rotor System with Transverse Breathing Crack and Initial Bending

Machines ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 79
Author(s):  
Yuehua Wang ◽  
Xin Xiong ◽  
Xiong Hu
Keyword(s):  
Nonlinear Response ◽  
Crack Depth ◽  
Rotor System ◽  
Frequency Change ◽  
Breathing Crack ◽  
Cracked Rotor ◽  
Rotor Systems ◽  
Vibrational Characteristics ◽  
The Stability ◽  
Cracked Shaft

This paper focuses on the stability and nonlinear response of a bearing-rotor system affected by a transverse crack and initial bending which was thought to be part of an unbalance or had been neglected before. The differences of breathing functions for the transverse breathing crack caused by initial bending is presented here, and the calculation of time-varying finite elements stiffness matrix of the cracked shaft is improved by replacing traditional the approximate crack segment with an exact area. After establishing the dynamic model of the cracked rotor with initial bending, vibrational characteristics such as amplitude-speed diagram, frequency spectrogram and bifurcations are investigated in detail. The eigenvalues of the transition matrix are calculated and analyzed as an indicator of dynamic stability with the growths of crack depth and initial bending. Many differences are found between the two cases of dynamic response of rotor system by numerical simulation. The frequency change with the growth of initial bending is opposite to the change with the growth of crack depth, and the shapes of amplitude-speed also having great different features. Stable regions are reduced and extended laterally by initial bending. All these results obtained in this paper will contribute to identify the bending fault and assess the stability of the bearing-rotor systems.

The Application of the Ritz Averaging Method to Determining the Response of Systems with Time Varying Stiffness to Harmonic Excitation

1980 ◽  
Vol 102 (2) ◽  
pp. 384-390 ◽  
Author(s):  
M. Benton ◽  
A. Seireg
Keyword(s):  
Linear System ◽  
Averaging Method ◽  
Approximate Solutions ◽  
Harmonic Excitation ◽  
Time Varying ◽  
Parametric Vibrations ◽  
Nonlinear Characteristics ◽  
Closed Form Solutions ◽  
Mathieu’S Equation ◽  
Stiffness Variation

Parametric vibrations occur in many mechanical systems such as gears where the stiffness variation and external excitations generally occur at integer multiples of the rotational speed. This paper describes a procedure based on the Ritz Averaging Method for developing closed form solutions for the response of such systems to harmonic excitations. Although the method is illustrated in the paper by the case of a linear system with harmonic stiffness fluctuation (defined by Mathieu’s equation) it can be readily applied to determine approximate solutions for systems with nonlinear characteristics and any periodic variations of parameters.

scholarly journals Dynamic Response of a Cracked Rotor With Some Comments on Crack Detection

1994 ◽  
Author(s):  
G. Meng ◽  
Eric J. Hahn
Keyword(s):  
Dynamic Response ◽  
Crack Detection ◽  
Harmonic Component ◽  
Major Axis ◽  
Vertical Axis ◽  
Crack Size ◽  
Cracked Rotor ◽  
Response Component ◽  
Crack Location ◽  
Backward Whirl

By considering time dependent terms as external excitation forces, the approximate dynamic response of a cracked horizontal rotor is analysed theoretically and numerically. The solution is good for small cracks and small vibrations in the stable operating range. For each steady state harmonic component the forward and backward whirl amplitudes, the shape and orientation of the elliptic orbit and the amplitude and phase of the response signals arc analysed, taking into account the effect of crack size, crack location, rotor speed and unbalance. It is found that the crack causes backward whirl, the amplitude of which increases with the crack. For a cracked rotor, the response orbit for each harmonic component is an ellipse, the shape and orientation of which depends on the crack size. The influence of the crack on the synchronous response of the system can be regarded as an additional unbalance whereupon, depending on the speed and the crack location, the response amplitude differs from that of the uncracked rotor. The nonsynchronous response provides evidence of crack in the sub-critical range, but is too small to be detected in the supercritical range. Possibilities for crack detection over the full speed range include the additional average (the constant) response component, the backward whirl of the response, the ellipticity of the orbit, the angle between the major axis and the vertical axis and the phase angle difference between vertical and horizontal vibration signals.

scholarly journals On Peculiarities of Betatron Oscillations of Accelerated Electron Bunches in Capillary Waveguides

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Mikhail Veysman
Keyword(s):  
Synchrotron Radiation ◽  
Parametric Excitation ◽  
Electron Bunch ◽  
Short Pulse ◽  
Spectral Parameters ◽  
Electron Bunches ◽  
Narrow Capillary ◽  
Parametric Resonances ◽  
Parametric Instabilities ◽  
The One

It is shown that the dynamics of electrons accelerated in narrow capillary waveguides is significantly influenced by the parametric excitation of their betatron oscillations. On the one hand, this excitation can irreversibly spoil the emittance of an accelerated electron bunch that limits the possibilities of their practical use. On the other hand, controlled parametric excitation of betatron oscillations can be used to generate short-pulse sources of synchrotron radiation. The article analyzes the regions of parametric instabilities, their dependence on the parameters of accelerated electron bunches and guiding structures, and their influence on the dynamics of accelerated electrons. The parameters of the generated synchrotron radiation are also estimated. Measurements of the spectral parameters of synchrotron radiation can serve as a tool for diagnostics of betatron oscillations and their excitation in the case of parametric resonances.

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