Transverse Open Crack Research Articles

NATURAL FREQUENCIES OF A NON-UNIFORM BEAM WITH MULTIPLE CRACKS VIA MODIFIED FOURIER SERIES

A new method is presented in this paper for computing the natural frequencies of a non-uniform beam with an arbitrary number of transverse open cracks. The essence of this new method lies in the use of a kind of modified Fourier series (MFS) which is specially developed for a beam with transverse open cracks. Unlike conventional Fourier series, modified Fourier series can approach a function with internal geometrical discontinuities. Based on the modified Fourier series, one can treat the cracked beam in the most usual way and thus reduce the problem to a simple one. The beam can be of non-uniform cross section and the number of cracks can be arbitrary. By using the present method, only standard linear eigenvalue equations, rather than non-linear algebraic equations, need to be solved. All the formulae are expressed in matrix form which renders the programming quite straightforward.

NATURAL FREQUENCIES OF A BEAM WITH AN ARBITRARY NUMBER OF CRACKS

In this article a new technique is proposed for calculating natural frequencies of a vibrating beam with an arbitrary finite number of transverse open cracks. The main feature of this method is related to decreasing the dimension of the matrix involved in the calculation, so that reduced computation time is required for evaluating natural frequencies compared to alternative methods which also make use of a continuous model of the beam.

Natural Vibrations of a Clamped-Clamped Arch With an Open Transverse Crack

The paper presents a finite element model of the arch with a transverse, one-edge crack. A part of the cracked arch is modelled by a curved beam finite element with the crack. Parts of the arch without the crack are modelled by noncracked curved beam finite elements. The crack occurring in the arch is nonpropagating and open. It is assumed that the crack changes only the stiffness of the arch, whereas the mass is unchanged. The method of the formation of the stiffness matrix of a curved beam finite element with the crack is presented. The effects of the crack location and its length on the changes of the in-plane natural frequencies and mode shapes of the clamped-clamped arch are studied.

VIBRATION ANALYSIS AND DIAGNOSIS OF A CRACKED SHAFT

A diagnostic method of determining the position and size of a transverse open crack on a stationary shaft without disengaging it from the machine system is investigated. The crack is modelled as a joint of a local spring. To obtain the dynamic characteristics of a stepped shaft and a multi-disc shaft, the transfer matrix method is employed on the basis of Timoshenko beam theory. It can be combined with beam segments to derive the frequency equation for the assembly, and is then solved for the frequency as well as the corresponding mode shape of the cracked shaft. Verification of this approach by comparison with some already existing published experimental data is presented. The position of the crack can be predicted by comparing the fundamental mode shapes of the shaft with and without a crack. Furthermore the depth of the crack can be obtained by the change of natural frequency of the shaft with and without a crack.

Modelling and vibration analysis of a cantilever composite beam with a transverse open crack

Abstract Eigenfrequencies of a cantilever beam, made from graphite-fiber reinforced polyimide, with a transverse on-edge non-propagating open crack are investigated. Two models of the beam are presented. In the first model the crack is modelled by a massless substitute spring. The flexibility of the spring is calculated on the basis of fracture mechanics and the Castigliano theorem. The second model is based on the finite element method (FEM). The undamaged parts of the beam are modelled by beam finite elements with three nodes and three degrees of freedom at the node. The damaged part of the beam is replaced by the cracked beam finite element with degrees of freedom identical to those of the non-cracked one. The effects of various parameters (the crack location, the crack depth, the volume fraction of fibers and the fibers’ orientation) upon the changes of the natural frequencies of the beam are studied. Computation results indicate that the decrease of the natural frequencies not only depends on the position of the crack and its depth, as in the case of isotropic material, but also that these changes strongly depend on the volume fraction of the fibers and the angle of the fibers of the composite material.

Natural Vibrations of Cracked Shafts Carrying Elastically Mounted End Masses

The effects of a transverse open crack on the modal frequency parameters of stationary shafts carrying elastically mounted end masses are presented. Dimarogonas' crack model has been considered in the problem formulation. This is a 2 × 2 local flexibility matrix with coupling terms. Most of the shaft configurations with classical and non-classical support conditions can be studied using the model presented. The modal frequency parameters for different design configurations and crack locations have been computed in order to clarify the vibrational behaviour and to specify the locations in which the crack becomes dangerous from the dynamic design point of view. Modal frequency graphs are provided for some engineering applications which may be of immediate use in design as identification tools for crack diagnosis and analysis.

Analysis of Progressive Matrix Cracking in Composite Laminates I. Thermoelastic Properties of a Ply with Cracks

Overall stiffnesses and compliances, thermal expansion coefficients, and stress and strain averages are evaluated for a fibrous composite lamina which contains a given density of open transverse cracks and is subjected to uniform mechanical loads and thermal changes. The evaluation procedure is based on the self-consistent method and is similar, in principle, to that used in finding elastic constants of unidirectional com posites.

On a factitious silvering or gilding, used in India

The dynamic stability of dynamical systems with time-periodic stiffness is addressed here. Cracked rotor systems with time-periodic stiffness are well-known examples of such systems. Time-varying area moments of inertia at the cracked element cross-section of a cracked rotor have been used to formulate the time-periodic finite element stiffness matrix. The semi-infinite coefficient matrix obtained by applying the harmonic balance (HB) solution to the finite element (FE) equations of motion is employed here to study the dynamic stability of the system. Consequently, the sign of the determinant of a scaled version of a sub-matrix of this semi-infinite coefficient matrix at a finite number of harmonics in the HB solution is found to be sufficient for identifying the major unstable zones of the system in the parameter plane. Specifically, it is found that the negative determinant always corresponds to unstable zones in all of the systems considered. This approach is applied to a parametrically excited Mathieu’s equation, a two degree-of-freedom linear time-periodic dynamical system, a cracked Jeffcott rotor and a finite element model of the cracked rotor system. Compared to the corresponding results obtained by Floquet’s theory, the sign of the determinant of the scaled sub-matrix is found to be an efficient tool for identifying the major unstable zones of the linear time-periodic parametrically excited systems, especially large-scale FE systems. Moreover, it is found that the unstable zones for a FE cracked rotor with an open transverse crack model only appear at the backward whirl. The theoretical and experimental results have been found to agree well for verifying that the open crack model excites the backward whirl amplitudes at the critical backward whirling rotational speeds.