J.E.T. Penny

A combined genetic and eigensensitivity algorithm for the location of damage in structures

Abstract Genetic algorithms have been the subject of considerable interest in recent years, since they appear to provide a robust search procedure for solving difficult problems. Due to the way the genetic algorithm explores the region of interest it avoids getting stuck at a particular local minimum and locates the global optimum. The genetic algorithm is slow in execution and is best applied to difficult problems. This paper applies a genetic algorithm to the problem of damage detection using vibration data. The objective is to identify the position of one or more damage sites in a structure, and to estimate the extent of the damage at these sites. The genetic algorithm is used to optimize the discrete damage location variables. For a given damage location site or sites, a standard eigensensitivity method is used to optimize the damage extent. This two-level approach incorporates the advantages of both the genetic algorithm and the eigensensitivity methods. The method is demonstrated on a simulated beam example and an experimental plate example.

Crack Modeling for Structural Health Monitoring

There are a number of approaches to the modeling of cracks in beam structures reported in the literature, that fall into three main categories; local stiffness reduction, discrete spring models, and complex models in two or three dimensions. This paper compares the different approaches to crack modeling, and demonstrates that for structural health monitoring using low frequency vibration, simple models of crack flexibility based on beam elements are adequate. This paper also addresses the effect of the excitation for breathing cracks, where the beam stiffness is bilinear, depending on whether the crack is open or closed. Most structural health monitoring methods assume that the structure is behaving linearly, whereas in practice the response will be nonlinear to an extent that varies with the form of the excitation. This paper will demonstrate these effects for a simple beam structure.

Automatic choice of measurement locations for dynamic testing

This paper examines the problem of choosing an optimum set of measurement locations for experimental modal testing and suggests criteria whereby the suitability of the chosen locations can be assessed. Two methods of coordinate selection are used: one based on Guyan reduction and the other on the Fisher information matrix. Each begins with a detailed finite element model of the structure being tested. Both procedures reduce this model by one degree of freedom at a time until the number of degrees of freedom in the reduced model equals the number of measurement locations required. The choice of the eliminated coordinates is generally automatic, and the coordinates of the reduced model are those used for modal testing

Parameter subset selection in damage location

Methods to locate damage in structures, using a finite element model and low frequency measured vibration data, have attracted considerable interest. A large number of parameters are required to ensure that the damage location and mechanism may be modelled by at least one set of parameter values. Generally the identified parameter values are not unique and extra information must be incorporated into the identification. The finite element model of a damaged structure is likely to be in error at only a small number of sites. This is equivalent to requiring that only a subset of parameters are in error, and leads to the methods of subset selection. The standard method uses the sensitivity matrix based on the initial finite element model to choose the parameter subset. Many residuals used for damage location are nonlinear functions of the parameters, and this paper examines the relationship between the subset selection and the iteration required for the parameter estimation. Also measurements are often taken ...

Simplified Modelling of Rotor Cracks

There are a number of approaches reported in the literature for modelling cracks in shafts. Although two and three dimensional finite element models may be used, the most popular approaches are based on beam models. This paper considers a number of different approaches to modelling cracks in shafts of rotating machines. Of particular concern is the modelling of breathing cracks, which open and close due to the selfweight of the rotor, producing a parametric excitation. In the simplest model the shaft stiffness is bilinear, depending on whether the crack is open or closed. More complicated functions that relate the shaft stiffness to angular position have been proposed, and two possibilities will be considered. This paper will demonstrate the influence of the crack model on the response of a Jeffcott rotor, using the harmonic balance approach.

Updating model parameters from frequency domain data via reduced order models

Abstract Methods to update the parameters of finite element models using measured vibration data usually use the experimentally derived modal model, that is the system natural frequencies, damping coefficients and mode shapes. The frequency response functions have been used directly to update condensed analytical models and so avoid the sometimes difficult step of deriving the modal model. This paper suggests algorithms to update selected physical parameters of a full finite element model using the frequency response functions. These algorithms are tested on a simulated example.

The accuracy and stability of time domain finite element solutions

Abstract The use of finite elements in the time domain provides a means of determining the response of a mechanical system to any forcing function. Two types of elements are used; a cubic element which maintains continuity of displacement and velocity between adjacent elements, and a quintic element which also ensures continuity of acceleration. The accuracy of solutions depends on the number of elements per unit time, errors being inversely proportional to the square of the number of elements for the cubic and inversely proportional to the fourth power of the number of elements for the quintic element. A condition for the stability of a solution is also established.

An integral equation approach to the fundamental frequency of vibrating beams

Abstract An approximation to the lowest natural frequency of vibrating beams is obtained analytically by applying eigenvalue, eigenfunction theory to the defining integral equation. The method produces successively closer values for both upper and lower bounds to the fundamental frequency. It is found that the second lower bound provides in itself a good approximation to published values and a graph is derived which provides a bound for the error in this approximation without further computation. The application of integral equations to the formulation of mechanical engineering problems is increasing and one aim of the paper is to draw attention to the possibility of obtaining analytical solutions.

Quantifying the Correlation Between Measured and Computed Mode Shapes

This paper proposes a method whereby a set of computed mode shapes for a structure and set of measured mode shapes may be compared through the mass matrix. The errors between the two are found to fall into three categories: (a) mutual orthogonality of the measured modes is not satisfied, (b) the sets of modal vectors do not span the same subspaces, and (c) the modes are not perfectly aligned with the common subspace. In all three cases, the errors emerge as a set of angles, and the number of angles associated with each class of error is the same as the number of modes in the sets being compared. The sets of angles can each be combined into a single error angle for each class and ultimately a single angle, which reflects the degree of agreement between the measured and computed mode shapes.

The Effect of Close or Repeated Eigenvalues on the Updating of Model Parameters from FRF Data

Methods to update the parameters of finite element models using measured vibration data usually use the experimentally derived modal model, that is, the system natural frequencies, damping coefficients, and mode shapes. Alternatively the frequency response functions have been used directly to update condensed analytical models and so avoid the sometimes difficult step of deriving the modal model. Previously the authors suggested an algorithm using FRF data that is basically a weighted equation error method based on a reduced order model. This paper investigates the performance of the algorithm for systems with closely coupled or repeated natural frequencies or eigenvalues.