Andrew D. Dimarogonas

Vibration of cracked structures: A state of the art review

Abstract The presence of a crack in a structural member introduces a local flexibility that affects its vibration response. Moreover, the crack will open and close in time depending on the rotation and vibration amplitude. In this case the system is nonlinear. Furthermore, if general motion is considered, the local stiffness matrix description of the cracked section of the shaft leads to a coupled system, while for an uncracked shaft the system is decoupled. This means that the crack introduces new harmonics in the spectrum. In fact, in addition to the second harmonic of rotation and the subharmonic of the critical speed, two more families of harmonics are observed: 1. (1) higher harmonics of the rotating speed due to the nonlinearity of the closing crack, and 2. (2) longitudinal and torsional harmonics are present in the start-up lateral vibration spectrum due to the coupling. Over 500 papers on the subject were published in the past 10 yrs. A wealth of analytical, numerical and experimental investigations now exists. However, a consistent cracked bar vibration theory is yet to be developed. There are still many unanswered questions, especially in the area of closing cracks in rotating shafts.

Identification of crack location and magnitude in a cantilever beam from the vibration modes

Measurement of flexural vibrations of a cantilever beam with rectangular cross-section having a transverse surface crack extending uniformly along the width of the beam and analytical results are used to relate the measured vibration modes to the crack location and depth. From the measured amplitudes at two points of the structure vibrating at one of its natural modes, the respective vibration frequency and an analytical solution of the dynamic response, the crack location can be found and depth can be estimated with satisfactory accuracy. The method can be used to identify cracks in structures by measuring its modal characteristics. It is a non-destructive testing method for crack identification, and it is applicable to structures for which a structural analysis is available. The main features of the method are as follows: (a) it requires amplitude measurements at two positions of the structure only; (b) it is applicable to all one-dimensional structures; (c) it demands modest computational effort; (d) it is an accurate, simple and easy to handle method having the advantage that the measurements may be carried out in situ with rather simple equipment.

Coupled longitudinal and bending vibrations of a rotating shaft with an open crack

Abstract The coupling of longitudinal and bending vibrations of a rotating shaft, due to an open transverse surface crack is investigated. The assumption of the open crack leads to a system with behaviour similar to that of a rotor with dissimilar moments of inertia along two perpendicular directions. The local flexibility due to the presence of the crack can be represented by way of a 6×6 matrix for six degrees of freedom in a short shaft element which includes the crack. This matrix has off-diagonal terms which cause coupling along the directions which are indicated by these terms. Here shear is not considered and three degrees of freedom are used: bending in the two main directions and extension. This leads to a 3×3 stiffness matrix with coupling terms. The undamped free and forced coupled vibration are first considered. The coupling is investigated and the effects of unbalance and gravity are examined. Then damped coupled vibration is considered for free and forced vibration. The existence of coupling between longitudinal and bending vibration due to the crack is a very useful property which, together with the sub-critical resonance due to crack, can form a basis for crack identification in rotating shafts. New and interesting phenomena of coupled transverse and longitudinal motion are presented and discussed.

Identification of cracks in welded joints of complex structures

Abstract The problem of the influence of a crack in a welded joint on the dynamic behaviour of a structural member is discussed in this work. Analytical and experimental investigation gave the relation between the change in natural frequency of vibration of a cantilever beam and the crack depth that appears at the built-in edge which is clamped by way of a weld. Measurement of the change of the natural frequency of the beam thus can give information for crack appearance and furthermore make possible the estimation of its depth. The results have been used to determine the behaviour of beams with different boundary conditions and can be extended to formed structures with complicated geometries.

Vibration of cracked shafts in bending

Abstract Cracked rotating shafts exhibit a certain particular dynamic response due to the local flexibility of the cracked section. In this response, most of the features of the response of a shaft with dissimilar moments of inertia can be identified. Moreover, the non-linear behavior of the closing crack introduces the characteristics of non-linear systems. For many practical applications, the system can be considered bi-linear and analytical methods can be applied. A de Laval rotor with an open crack is investigated by way of application of the theory of shafts with dissimilar moments of inertia. Furthermore, analytical solutions are obtained for the closing crack under the assumption of large static deflections, a situation common in turbomachinery. Finally, a solution is developed for the case in which the local flexibility function is found experimentally.

Stability of columns with a single crack subjected to follower and vertical loads

Abstract A general flexibility matrix is developed which expresses the local flexibility of a beam of rectangular cross section with a single edge crack. The dominant term in this matrix is used to study the stability of the cracked column to follower and vertical loads. The kinetic method is used and eigencurves are developed to study the system stability. For follower type loads, the Beck column, stability charts are given for several crack locations and sizes. Flutter type of instability is always encountered. For vertical loads the same type of analysis reveals the divergent type of instability, reported already in the literature. It was found that cracks can make this system have flutter type of instability for vertical loads.

Coupling of bending and torsional vibration of a cracked Timoshenko shaft

SummaryA transverse surface crack is known to add to the shaft a local flexibility due to the stress-strain singularity in the vicinity of the crack tip. This flexibility can be represented by way of a 6 × 6 matrix describing the local flexibility in a short shaft element which includes the crack. This matrix has off-diagonal terms which cause coupling of motion along the directions which are indicated by the off-diagonal terms. Not all motions are coupled, however. To study the coupling of torsion and shear, a 3 × 3 flexibility matrix is used which includes the appropriate terms. Due to the shear terms of the Timoshenko beam equation of the shaft, bending vibration is finally coupled to torsional vibration. This effect is the subject of this investigation, which is of particular importance in turbomachinery operation. The equations of motion of a Timoshenko beam shaft with three degrees of freedom are derived. The free vibration of the shaft and the influence of the crack on the vibrational behaviour of the shaft is studied. The relation of the eigenvalues of the system, to the crack depth and the slenderness ratio of the shaft is derived. Moreover forced vibration analysis of the cracked shaft is performed. The significant influence of the bending vibration on the torsional vibration spectrum, and vice-versa, is demonstrated. It is believed that this effect can be very useful for rotor crack identification in service, which is of importance to turbomachinery.ÜbersichtBekanntlich verringert ein von der Oberfläche in den Querschnitt reichender Riß infolge der Spannungs- und Verzerrungssingularität an der Rißspitze örtlich die Steifigkeit einer Welle. Dies kann mit Hilfe einer 6 × 6-Nachgiebigkeitsmatrix für ein kurzes Wellenstück, das den Riß enthält, beschrieben werden. Die Matrix enthält Elemente außerhalb der Diagonalen, wodurch eine Kopplung der Bewegungen in die Richtungen erfolgt, welche das Element anzeigt. Nicht alle Bewegungen sind dabei verknüpft. Zur Untersuchung der Kopplung von Torsion und Querschub wird eine 3 × 3-Nachgiebigkeitsmatrix benutzt, die die betreffenden Elemente enthält. Infolge der Querkraft-Terme in der Timoshenko-Balkengleichung werden letztendlich Biegef- und Torsionsschwingungen gekoppelt. Dieser Effekt ist Gegenstand der Untersuchung. Die Bewegungsgleichungen eines Timoshenko-Balkens mit 3 Freiheitsgraden werden hergeleitet. Die freien Schwingungen der Welle und der Einfluß des Risses auf das Schwingungsverhalten werden untersucht. Die Beziehungen zwischen Eigenformen, Rißtiefe und Schlankheitsgrad der Welle werden hergeleitet. Darüber hinaus werden erzwungene Schwingungen der angerissenen Welle untersucht. Der deutliche Einfluß der Biegeschwingung auf das Spektrum der Torsionsschwingung und umgekehrt wird aufgezeigt. Dieser Effekt ist bei Turbomaschinen bedeutsam, da er für die Identifizierung von Rotorrissen beim Betrieb nützlich sein müßte.

Stability of cracked rotors in the coupled vibration mode

A transverse surface crack is known to add to a shaft a local flexibility due to the stress-strain singularity in the vicinity of the crack tip. This flexibility can be represented, in the general case by way of a 6 × 6 compliance matrix describing the local flexibility in a short shaft element which includes the crack. This matrix has off-diagonal terms which cause coupling along the directions which are indicated by the off-diagonal terms. In addition, when the shaft rotates the crack opens and closes. Then the differential equations of motion have periodically varying stiffness coefficients and the solution can be expressed as a sum of harmonic functions of time. A method for the determination of the intervals of instability of the first and of second kind is developed. The results have been presented in stability charts in the frequency vs. depth of the crack domain. The coupling effect due to the crack leads to very interesting results such as new frequencies and vibration modes.

Coupled Longitudinal and Bending Vibrations of a Cracked Shaft

This paper describes the coupling of bending and longitudinal vibration of a stationary cracked shaft with an open crack. The crack is modeled by way of a 2×2 local flexibility matrix with coupling terms. The elements of this matrix are obtained analytically. One of the elements compares well with experimental data of other investigators. The free vibration of the shaft, and the influence of the crack on the vibrational behavior of the shaft is studied. The relation of the eigenvalues of the system and the crack depth as functions of the slenderness ratio are presented. The forced vibration of the shaft is also studied and the coupling of the vibration modes is verified analytically and experimentally. The applicability of the method as a crack identification tool is demonstrated.

Dynamic Sensitivity of Structures to Cracks

Cracks that develop on machine members and structures influence their dynamic behavior. The Rayleigh principle is used for an estimation of the change in the natural frequencies and modes of vibation of the structure if the crack geometry is known, assuming that the eigenvalue problem for the uncracked structure has been solved in advance. The method reduces the computational effort needed for the full eigensolution of cracked structures and gives acceptable accuracy. It can be extended to higher modes and to decompose degenerate modes found in symmetric structures. To demonstrate the change in the dynamic behavior of linear structures with the crack depth, a cylindrical shaft and a plane frame consisting of prismatic bars were analyzed for dynamic sensitivity to surface cracks.