Development of a Novel Algorithm for a Crack Detection, Localization, and Sizing in a Beam Based on Forced Response Measurements

2008 ◽  
Vol 130 (2) ◽  
Author(s):  
M. Karthikeyan ◽  
R. Tiwari ◽  
S. Talukdar
Keyword(s):  
Natural Frequency ◽  
Crack Detection ◽  
Beam Theory ◽  
Crack Size ◽  
Forced Response ◽  
Crack Model ◽  
Beam Model ◽  
Crack Location ◽  
Forced Responses ◽  
Crack Element

The present work aims at the development of a method for the crack detection, localization and sizing in a beam based on the transverse force and response signals. The Timoshenko beam theory is applied for transverse vibrations of the beam model. The finite element method is used for the cracked beam forced vibration analysis. An open transverse surface crack is considered for the crack model, which contains standard five flexibility coefficients. The effect of the proportionate damping is also included. A harmonic force of known amplitude with sine-sweep frequency is used to dynamically excite the beam, up to few flexible modes, which could be provided with the help of an exciter. In practice, linear degrees of freedom (DOFs) can be measured quite accurately; however, rotational DOFs are difficult to measure accurately. All rotational DOFs, except at crack element, are eliminated by a dynamic condensation scheme; for elimination of rotational DOFs at the crack element, a new condensation scheme is implemented. The algorithm is iterative in nature and starts with a presumption that a crack is present in the beam. For an assumed crack location, flexibility coefficients are estimated with the help of forced responses. The Tikhonov regularization technique is applied in the estimation of bounded crack flexibility coefficients. These crack flexibility coefficients are used to obtain the crack size by minimizing an objective function. With the help of the estimated crack size and measured natural frequency, the crack location is updated. The procedure iterates till the crack size and location get stabilized up to the desired level of accuracy. The algorithm has a potential to detect no crack condition also. The crack flexibility and damping coefficients are estimated as a by-product. Numerical examples, with the simply supported and cantilevered beams, are given to justify the applicability and versatility of the algorithm in practice. With the numerically simulated forced responses, which have the noise contamination and the error in the natural frequency measurements, the estimated crack parameters (i.e., the crack location and size) are in good agreement.

Rotordynamic Crack Diagnosis: Distinguishing Crack Depth and Location

2013 ◽  
Author(s):  
Philip Varney ◽  
Itzhak Green
Keyword(s):  
Fatigue Crack ◽  
Condition Monitoring ◽  
Crack Depth ◽  
Forced Response ◽  
System Response ◽  
Contour Plot ◽  
Crack Model ◽  
Finite Width ◽  
Angular Response ◽  
Crack Location

The goal of this work is to establish a condition monitoring regimen capable of diagnosing the depth and location of a transverse fatigue crack in a rotordynamic system. The success of an on-line crack diagnosis regimen hinges on the accuracy of the crack model used. The model should account for the depth of the crack and the localization of the crack along the shaft. Negating the influence of crack location on system response ignores a crucial component of real cracks. Two gaping crack models are presented; the first simulates a finite-width manufactured notch, while the second models an open fatigue crack. An overhung rotordynamic system is modeled, imitating an available rotordynamic test rig. Four degree-of-freedom equations of motion for both crack models are presented and discussed, along with corresponding transfer matrix techniques. Free and forced response analyses are performed, with emphasis placed on results applicable to condition monitoring. It is demonstrated that two identifiers are necessary to diagnose the crack parameters: the 2X resonance frequency and the magnitude of the 2X component of the rotor angular response at resonance. First, a contour plot of the 2X resonant shaft speed versus crack depth and location is generated. The magnitude of the 2X component of the rotor’s angular response along the desired contour is obtained, narrowing the possible pairs of crack location/depth to either one or two possibilities. Practical aspects of the diagnosis procedure are then discussed.

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.

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

1997 ◽  
Vol 119 (2) ◽  
pp. 447-455 ◽  
Author(s):  
G. Meng ◽  
E. 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 analyzed 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 are analyzed, 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 depend 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 subcritical 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.

The effect of crack geometry on non-destructive fault detection of EN 8 and EN 47 cracked cantilever beam

2019 ◽  
Vol 50 (3) ◽  
pp. 92-100 ◽  
Author(s):  
V Khalkar ◽  
S Ramachandran
Keyword(s):  
Free Vibration ◽  
Natural Frequency ◽  
Cantilever Beam ◽  
Crack Detection ◽  
Crack Depth ◽  
Free Vibration Analysis ◽  
Crack Location ◽  
Crack Geometry ◽  
Shaped Crack ◽  
Non Destructive

Since long it has been observed that the size of the crack in structures increases with time, and finally, it may lead to its catastrophic failure. Hence, it is crucial to do the vibration study of cracked structures with regard to vibration-based crack detection and the classification of cracks. So far, vibration-based non-destructive testing method is applied to many spring steel cracked cantilever beams for its possible crack detection. However, the effect of various kinds of practical cracks, that is, V-shaped and U-shaped, on the applicability of these methods has been overlooked. To investigate this issue, artificially cracks are made on the cantilever beam. By free vibration analysis, the effect of crack geometry, crack depth, and crack location on natural frequency is investigated. The natural frequency results obtained from V-shaped and U-shaped models for the same crack configurations are compared with each other and it is revealed that the results are not much sensitive for the change of crack geometry. Hence, it is clear that free vibration-based crack detection method approximately predicts the crack parameters, that is, crack location and crack depth, in structures irrespective of the crack geometry. It is also found that for the same configuration, results of natural frequency are comparatively on the lower side for U-shaped crack models than V-shaped crack models. In this study, the natural frequency of each cracked case is computed by a theoretical method and numerical method and shows good agreement. Finally, it is also observed that structural integrity of a cracked cantilever beam is a function of crack location.

scholarly journals Effects of Crack on Vibration Characteristics of Mistuned Rotated Blades

2017 ◽  
Vol 2017 ◽  
pp. 1-18 ◽  
Author(s):  
Hailong Xu ◽  
Zhongsheng Chen ◽  
Yongmin Yang ◽  
Limin Tao ◽  
Xuefeng Chen
Keyword(s):  
Natural Frequency ◽  
Vibration Amplitude ◽  
Crack Detection ◽  
Vibration Characteristics ◽  
Lumped Parameter Model ◽  
Crack Model ◽  
Breathing Crack ◽  
Vibration Localization ◽  
Lumped Parameter ◽  
Amplitude Increase

Rotated blades are key mechanical components in turbine and high cycle fatigues often induce blade cracks. Meanwhile, mistuning is inevitable in rotated blades, which often makes it much difficult to detect cracks. In order to solve this problem, it is important and necessary to study effects of crack on vibration characteristics of mistuned rotated blades (MRBs). Firstly, a lumped-parameter model is established based on coupled multiple blades, where mistuned stiffness with normal distribution is introduced. Next, a breathing crack model is adopted and eigenvalue analysis is used in coupled lumped-parameter model. Then, numerical analysis is done and effects of depths and positions of a crack on natural frequency, vibration amplitude, and vibration localization parameters are studied. The results show that a crack causes natural frequency decease and vibration amplitude increase of cracked blade. Bifurcations will occur due to a breathing crack. Furthermore, based on natural frequencies and vibration amplitudes, variational factors are defined to detect a crack in MRBs, which are validated by numerical simulations. Thus, the proposed method provides theoretical guidance for crack detection in MRBs.

Dynamic Response of Tubular T-Joints Under the Influence of Propagating Cracks

1996 ◽  
Vol 118 (1) ◽  
pp. 71-78 ◽  
Author(s):  
D. I. Nwosu ◽  
A. S. J. Swamidas ◽  
J. Y. Guigne´
Keyword(s):  
Natural Frequency ◽  
Crack Detection ◽  
Crack Depth ◽  
Bending Moment ◽  
Frequency Change ◽  
T Joints ◽  
Finite Element Program ◽  
Crack Location ◽  
Percent Change ◽  
Frequency Changes

This paper presents an analytical study on the vibration response of tubular T-joints for detecting the existence of cracks along their intersections. The ABAQUS finite element program was utilized for carrying out the analysis. Frequency response functions were obtained for a joint with and without cracks. The joint was modeled with 8-node degenerate shell elements having 5 degrees of freedom per node. Line spring elements were used to model the crack. The exact crack configuration (semielliptical shape, Fig. 5(b)), as observed from numerous experimental fatigue crack investigations at the critical location, has been achieved through a mapping function, that allows a crack in a planar element to be mapped on to the tube surface. The natural frequency changes with respect to crack depth show little changes, being 4.82 percent for a 83-percent crack depth for the first mode. On the other hand, significant changes have been observed for bending moment and curvature as a function of crack depth. For an 83-percent chord thickness crack, a 97-percent change in bending moment at points around the crack vicinity, and 34.15 to 78 percent change in bending moments, for those locations far away from the crack location, have been observed. Natural frequency change should be combined with other modal parameters such as “bending moment (or bending strain)” and “curvature” changes for crack detection. The presence of the crack can be detected at locations far away from the crack location using such sensors as strain gages.

Rotordynamic Crack Diagnosis: Distinguishing Crack Depth and Location

2013 ◽  
Vol 135 (11) ◽  
Author(s):  
Philip Varney ◽  
Itzhak Green
Keyword(s):  
Condition Monitoring ◽  
Crack Detection ◽  
Fatigue Cracks ◽  
Equations Of Motion ◽  
Crack Depth ◽  
Forced Response ◽  
Contour Plot ◽  
Crack Model ◽  
Finite Width ◽  
Shaft Speed

The goal of this work is to establish simple condition monitoring principles for diagnosing the depth and location of transverse fatigue cracks in a rotordynamic system. The success of an on-line crack diagnosis regimen hinges on the accuracy of the crack model, which should account for the crack's depth and location. Two gaping crack models are presented; the first emulates a finite-width notch typically manufactured for experimental purposes, while the second models a gaping fatigue crack. The rotordynamic model used herein is based upon an available overhung rotordynamic test rig that was originally constructed to monitor the dynamics of a mechanical face seal. Four degree-of-freedom, linear equations of motion for both crack models are presented and discussed. Free and forced response analyses are presented, emphasizing results applicable to condition monitoring and, particularly, to diagnosing the crack parameters. The results demonstrate that two identifiers are required to diagnose the crack parameters: the 2X resonance shaft speed and the magnitude of the angular 2X subharmonic resonance. First, a contour plot of the 2X resonance shaft speed versus crack depth and location is generated. The magnitude of the 2X resonance along the desired 2X frequency contour is then obtained, narrowing the possible pairs of crack location and depth to either one or two possibilities. Practical aspects of the suggested diagnostic procedure are discussed, as well as qualitative observations concerning crack detection.

Dynamics of Large Power Rotating Machine with Cracked Shaft

2005 ◽  
Vol 293-294 ◽  
pp. 337-346
Author(s):  
Slawomir Banaszek
Keyword(s):  
Crack Detection ◽  
Crack Depth ◽  
Computer Code ◽  
Crack Model ◽  
Simulation Research ◽  
Large Power ◽  
Crack Location ◽  
Rotating Machine ◽  
Diagnostic Indicator ◽  
Cracked Shaft

The paper presents the course and results of crack propagation simulation research. The object taken into account is a large power turbo-set rotor. The computer code system NLDW is presented. It uses a non-linear model of journal bearings, and well known crack model. Crack depth is marked by a crack coefficient. It is shown the crack generates a coupled forms of lateral, axial and torsional vibrations in multi-support rotor. Their intensity depends on the axial and circumferential crack location on the shaft. The attempt at pointing a proper diagnostic indicator for crack detection in large rotating machine is made according to obtained results.

Approximate analytical method for damage detection in free–free beam by measurement of axial vibrations

2012 ◽  
Vol 22 (1) ◽  
pp. 133-142 ◽  
Author(s):  
FB Sayyad ◽  
B Kumar ◽  
SA Khan
Keyword(s):  
Crack Detection ◽  
Natural Frequencies ◽  
Preventive Measure ◽  
Theoretical Method ◽  
Crack Size ◽  
Structural Element ◽  
Global Parameter ◽  
Crack Location ◽  
Vibration Data ◽  
The Impact

Nowadays, sophisticated structures and machinery parts are constructed by using metallic beams. Beams are widely used as structural element in civil, mechanical, naval, and aeronautical engineering. In structures and machinery, one undesirable phenomenon is crack initiation in which the impact cannot be seen overnight. Cracks develop gradually through time that lead finally to catastrophic failure. Therefore, crack should be monitored regularly with more care. This will lead to more effective preventive measure and ensure continuous operation of the structure and machine. Damage in structure alters its dynamic characteristics. The change is characterized by change in modal parameters, that is, modal frequencies. Thus, vibration technique can be suitably used as a nondestructive test for crack detection of component to be tested. Mostly modal frequencies are used for monitoring the crack because modal frequencies are properties of the whole structure component. In this paper, efforts are made to develop suitable methods that can serve as the basis to detection of crack location and crack size from measured axial vibration data. This method is used to address the inverse problem of assessing the crack location and crack size in various beam structure. The method is based on measurement of axial natural frequencies, which are global parameter and can be easily measured from any point on the structure and also indeed, the advantage in modeling complexity. In theoretical analysis, the relationship between the natural frequencies, crack location, and crack size has been developed. For identification of crack location and crack size, it was shown that data on the variation of the first two natural frequencies is sufficient. The experimental analysis is done to verify the practical applicability of the theoretical method developed.

scholarly journals Beam Theory of Thermal–Electro-Mechanical Coupling for Single-Wall Carbon Nanotubes

Nanomaterials ◽  
2021 ◽  
Vol 11 (4) ◽  
pp. 923
Author(s):  
Kun Huang ◽  
Ji Yao
Keyword(s):  
Mechanical Properties ◽  
Carbon Nanotubes ◽  
Beam Theory ◽  
Bending Stiffness ◽  
Single Wall Carbon Nanotubes ◽  
Beam Model ◽  
Euler Beam ◽  
New Model ◽  
Mechanical Coupling ◽  
Electrostatic Fields

The potential application field of single-walled carbon nanotubes (SWCNTs) is immense, due to their remarkable mechanical and electrical properties. However, their mechanical properties under combined physical fields have not attracted researchers’ attention. For the first time, the present paper proposes beam theory to model SWCNTs’ mechanical properties under combined temperature and electrostatic fields. Unlike the classical Bernoulli–Euler beam model, this new model has independent extensional stiffness and bending stiffness. Static bending, buckling, and nonlinear vibrations are investigated through the classical beam model and the new model. The results show that the classical beam model significantly underestimates the influence of temperature and electrostatic fields on the mechanical properties of SWCNTs because the model overestimates the bending stiffness. The results also suggest that it may be necessary to re-examine the accuracy of the classical beam model of SWCNTs.

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