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

doi:10.1016/0022-460X(87)90437-8

Journal of Sound and Vibration

Volume 117, Issue 1, 22 August 1987, Pages 81-93

https://doi.org/10.1016/0022-460X(87)90437-8 Get rights and content

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.

References (25)

H. Okamura
A cracked column under compression
Engineering Fracture Mechanics
(1969)
H. Liebowitz et al.
Carrying capacity of notched columns
International Journal of Solids and Structures
(1967)
A.D. Dimarogonas et al.
Vibration of cracked shafts in bending
Journal of Sound and Vibration
(1983)
A.D. Dimarogonas et al.
Analytical Methods in Rotor Dynamics
(1983)
H. Liebowtiz et al.
Failure of notched columns
Engineering Fracture Mechanics
(1968)
J.R. Rice et al.
The part-through surface crack in elastic plate
Journal of Applied Mechanics
(1972)
A.D. Dimarogonas et al.
Torsional vibration of a shaft with a cricumferential crack
Engineering Fracture Mechanics
(1980)
H.J. Petroski
The permanent deformation of a cracked cantilever struck transversely at its tip
Journal of Applied Mechanics
(1984)
C.A. Papadopoulos et al.
Coupling of bending and torsional vibration of a cracked Timoshenko shaft
Ingenieur Archive
(1987)
A.J. Bush
Experimentally determined stress-intensity factors for single-edge-crack round bars loaded in bending
Experimental Mechanics
(1976)

J.A. Henry

Monitoring rotating machinery to detect the growth of rotor cracks

I.W. Mayes et al.

A method of calculating the vibrational behaviour of coupled rotainng shafts containing a transferse crack

Cited by (438)

Simulation of crack closure effect on cracked beam vibrations
2024, Engineering Fracture Mechanics
Vibrations of beams containing cracks are usually investigated by using open or breathing crack models. Neither one of these approaches considers the plasticity induced crack closure (PICC) phenomena which occurs in fatigue crack growth. In this study, PICC is integrated into the vibration analysis of a cracked beam. First, fatigue crack growth at a notch is simulated with elastoplastic finite element analysis. Residual deformations, stresses and the contact pressures on crack faces are obtained. In the second phase, the model which includes these quantities is used for forced vibration analysis under harmonic loading by using implicit finite element method. For comparison, analyses are also running without PICC. Simulation results indicate that PICC has a significant effect on vibrations of cracked beams. For low load amplitude levels, cracked beam behaves like an intact beam. Nonlinear behavior appears only at rather high amplitude levels. It is concluded that PICC could suppress the crack breathing behavior and this could make it difficult to detect a fatigue crack in a beam from its nonlinear vibration response.
Programmatic model updating of damaged structures using modified SERR method
2023, Computers and Structures
Vision inspection information of damages in civil engineering structures is hard to be transformed into an analysis-available resistance model to conduct in-service condition assessment. This paper proposes an updating crack-embedded finite element modeling framework using periodical vision inspection information, which offers a convenient crack embedding approach for safety assessment of cracked reinforced concrete (RC) frame structures. For this purpose, a modified stain energy release rate (SERR) method is proposed to enable the method for solid element with high accuracy. Consequently, an algorithm is proposed to map the crack information obtained by vision method in the FE model. The combination of modified SERR method and proposed mapping algorithm could achieve flexibility matrix updating of cracked element. In addition, the effectiveness and generality of the proposed framework are validated using three cracking configurations. This research can provide a feasible framework for the rapid condition assessment of in-service civil engineering structures, showing a great prospect of intelligent disaster prevention.
Dynamic modeling and vibration analysis of cracked rotor-bearing system based on rigid body element method
2023, Mechanical Systems and Signal Processing
Rotor crack occurs frequently in rotating machinery. It is of great significance to study the vibration mechanisms of the cracked rotor-bearing system, which can provide proofs for crack online monitoring. In this paper, a new dynamic model of the cracked rotor-bearing system is developed using the rigid body element (RBE) method, and vibration response analysis is then carried out. The rotor is discretized as a set of RBEs, and each two adjacent RBEs is connected by an elastic joint (EJ). Each EJ is modeled with six imaginary springs (three translational springs and three rotational springs). The breathing crack is modeled through time-varying stiffnesses of imaginary springs, which are obtained using real-time parameters (area, cross-section moment of inertia, and polar moment of inertia) in four processes (crack fully closed, crack partially opened, crack fully opened, and crack partially closed) in a full rotation angle. The interactions of adjacent RBEs are then given based on the force and moment analysis, and the bearing forces are obtained based on two models (i.e., equivalent spring-damper model, and Gupta bearing model). The equations of motions of the cracked rotor-bearing system are developed through Newton-Euler equations and solved by the fourth-order Runge-Kutta-Fehlberg method with a step-changing criterion. The displacement responses are then obtained, and the frequency spectrums of displacement responses and their instantaneous frequencies (IFs) under different rotation speeds are analyzed. Simulation results of the cracked rotor-bearing system are in good agreement with previously published results and experiment results. What’s more, analysis results show that the frequency spectrum of IF can be an indicator for online monitoring of the rotor crack.
Adaptive modeling of vibrations and structural fatigue for analyzing crack propagation in a rotating system
2022, Journal of Sound and Vibration
This study is concerned with adaptive modeling of vibrations and structural fatigue in a rotating system with a propagating crack. The system consists of a Jeffcott rotor with a transverse surface crack at mid-span. It undergoes low and high cycle loadings due to an unbalance force. The high cycle loading (HCL) and the consequent high cycle fatigue (HCF) can accelerate crack growth, structural degradation and occurrence of mechanical failure. Therefore, accurate modeling is critical for lifetime prediction and prognostics requirements. There is a significant difference between the time constants of the rotor’s dynamic behavior and that of crack growth. Hence, in most of the prior studies, it has been assumed that the crack depth is constant, which implies that the system undergoes low cycle loading (LCL). In some studies that do focus on HCL, the loading scenario is assumed to be a summation of LCL steps with constant length. This study shows that assuming constant and pre-defined length for these steps can significantly affect the accuracy of the model leading to inaccurate lifetime prediction and poor prognostics in general. Hence, in this study, a novel approach is proposed in which the LCL step length is variable and adaptive. Considering different relationships proposed for modeling the fatigue crack growth, fluctuation of stress intensity factor (SIF) is the most important factor for driving the crack growth. Hence, an index is defined based on the rate of SIF fluctuation. Critical thresholds of this index are set at rapid and slow crack growth conditions. The level of proximity to these thresholds determines each LCL step length, which is hence determined adaptively. The equations governing the rotor motion and crack growth are formulated using Newton’s second law of motion and Paris–Erdogan’s fatigue crack growth law, respectively. The equations are solved numerically using the Runge–Kutta method. It is shown that this novel adaptive approach makes the modeling process realistic and computationally efficient while providing high accuracy. The results of this study could be of significance in decision making processes that would require lifetime prediction and prognostics of crack growth.
Vibration of locally cracked pre-loaded parabolic arches
2022, Journal of Sound and Vibration
We study linear dynamics of an initially parabolic arch deformed by a uniform ‘dead’ load. The arch is seen as a fully deformable one-dimensional continuum with rigid cross-sections, one of which suffers from a small local crack at its boundary. The crack is simulated by springs, the stiffnesses of which are evaluated via stress intensity factors. By two first-order perturbations we investigate a non-trivial equilibrium adjacent to the reference configuration and small vibration superposed on it. The modulation of the initial load on the natural angular frequencies and its consequences on damage detection is described and commented. It turns out that neglecting the initial load, recalling for actual ‘dead’ structural actions, can be misleading in damage identification, while its inclusion leads to better results.
A variance-based approach for the detection and localization of cracks in a beam
2022, Structures
A variance-based method is proposed for the detection and localization of cracks in a beam. A crack in a beam introduces a slope discontinuity in the beam elastic line. The local slope discontinuity due to the crack results in a localized change in the beam deflection variance values. In the present work, the local variance is obtained using a rectangular moving window along the beam length. A significant change in the local variance values is observed at the crack location. A two-time application of moving window variance leads to a clear spike at the crack location. Localization of cracks in cantilever and simply supported beams is presented. The proposed method is compared with the widely used wavelet-based method for crack localization. The present method is more suitable for locating the crack in the high slope region of beam deflection. Finite element analysis is used to obtain the simulated cracked cantilever and simply supported beam responses. White Gaussian noise is added to the simulated beam deflection to test the noise-robustness of the algorithm. A key challenge in the detection of cracks in a beam is to obtain a high-resolution spatial beam deflection measurement. For experimental verification, the high-resolution spatial measurement of the vibrating beam is obtained by photographic measurement. The high spatial resolution gives localized spikes corresponding to the crack location. The proposed algorithm is capable of localizing multiple simultaneous cracks in a beam. Experimental localization of cracks in single and double cracked cantilever beams is presented.

View all citing articles on Scopus

^†: Now at the Department of Mechanical Engineering, Washington University, St. Louis, Missouri 63130, U.S.A.

View full text

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

Abstract

Engineering Fracture Mechanics

International Journal of Solids and Structures

Journal of Sound and Vibration

Analytical Methods in Rotor Dynamics

Failure of notched columns

Engineering Fracture Mechanics

The part-through surface crack in elastic plate

Journal of Applied Mechanics

Torsional vibration of a shaft with a cricumferential crack

Engineering Fracture Mechanics

The permanent deformation of a cracked cantilever struck transversely at its tip

Journal of Applied Mechanics

Coupling of bending and torsional vibration of a cracked Timoshenko shaft

Ingenieur Archive

Experimentally determined stress-intensity factors for single-edge-crack round bars loaded in bending

Experimental Mechanics

Monitoring rotating machinery to detect the growth of rotor cracks

A method of calculating the vibrational behaviour of coupled rotainng shafts containing a transferse crack