Is Damage Location using Vibration Measurements Practical?
EUROMECH 365 International Workshop: DAMAS 97, Structural Damage Assessment using Advanced Signal Processing Procedures, Sheffield, UK, June/July 1997
IS DAMAGE LOCATION USING VIBRATION MEASUREMENTS PRACTICAL?
M. I. Friswell
Department of Mechanical Engineering
University of Wales Swansea, UK
J. E. T. Penny
Department of Mechanical and Electrical Engineering
Aston University, UK
This paper considers the state of the art in damage detection and location using low frequency vibration data. The general classes of methods are briefly reviewed and the practical difficulties are highlighted. Some possible ways forward are presented.
The area of damage detection and location using measured low frequency vibration data has attracted considerable attention recently. Doebling et al.  have presented an extensive survey of the field which should be consulted for further details. The object of this paper is somewhat different. Although a summary of the types of method employed will be given, the object is to determine the practical limitations of these methods and to try to suggest a way forward for further work in this field. This paper is designed to carry on the debate into the application of methods using low frequency vibration data and to generate discussion. The opinions expressed are personnel and many of the issues are controversial and we have not shirked from giving an opinion on these matters.
REVIEW OF DAMAGE LOCATION METHODS
Although there are many papers on damage detection and location they may be categorised by the type of data that they use and the form of parameterisation that they identify. These categories of techniques will now be outlined.
The Measured Data
There are three basic types of data used in the measurement of dynamics; time domain, frequency domain and the modal model. During Experimental Modal Analysis the sampled time series data is processed into the frequency response function data. For ambient excitation the spectrum of the responses are calculated. This frequency data is then further processed by curve fitting to obtain the modal model, that is the natural frequencies, damping ratios and mode shapes. At each stage the processing involves some compression of the data, and results in a reduction in the volume of data. It may be thought best to use time series data as the number of data points is very high. For linear systems there is little loss of information going from the time domain to the frequency domain. Indeed there is the advantage that the data may be averaged easily and so the affect of random noise is reduced.
The modal model represents a further reduction in the number of data points. But is the quantity and quality of information reduced? The FRF may be reconstructed quite accurately using the modal model, and this is done to check the accuracy of the curve fitting. Some researchers believe that the FRF data contains some information from the out of range modes. While theoretically this might be true, in practice, unless the mode is just out of the range, the modes outside the range are easily corrupted by noise and the response is dominated by the in band modes. Thus the FRF and the modal model essentially contain the same information. The advantage of the modal model is that it may be checked to ensure that the data looks reasonable. The advantage of the FRF is that the data is closer to its raw state and the curve fitting may eliminate effects that would help in the identification of damage. For algorithms based on a predominantly linear model, this is unlikely to be an advantage. The equivalence of the FRF and modal data means that it should be possible to draw the same conclusions when presented with equivalent data. Problems arise mainly because the weights required for the data and the initial FE model to ensure that the identification is equivalent are difficult to determine.
It is commonly acknowledged that natural frequencies may be measured more accurately than mode shapes. Typical resolution for the natural frequencies of a lightly damped mechanical structure are 0.1% whereas typical mode shape errors are 10% or more. The situation is further complicated because mode shapes are relatively insensitive to damage. For example, if a cantilever beam contains a crack, the first bending mode will look very much like the first mode of the undamaged beam, until the damage is very severe. Thus natural frequencies should be weighted very heavily compared to mode shapes in any identification exercise. Mode shapes are still valuable to pair the analytical and experimental modes, and also for 'symmetrical' structures which often have areas which produce similar changes to the natural frequencies.
Updating Mass and Stiffness Matrices
Early papers in updating changed complete mass and/or stiffness matrices [2, 3]. The goal was to reproduce the measured data (usually the modal model), by changing the stiffness matrix as little as possible (in some minimum norm sense). A number of problems exist with these methods. There is no guarantee that the resulting matrices are positive definite (or semi-definite for structures with free-free modes), and extra modes may be introduced into the frequency range of interest. The standard methods do not enforce the connectivity of the structure, represented by the bandedness of the matrices and the pattern of zero terms. Kabé gave a method that enforced the expected connectivity. More fundamental is that forcing the model to reproduce the data doesn't allow for the errors that will be present in the measured data. Mode shapes, in particular, can only be measured with a limited accuracy. Further errors are introduced when the mode shapes are expanded, as they need to be since very few of the analytical degrees of freedom will be measured.
The major problem for damage location, and indeed for error location in model updating, is that all elements in the matrices may be changed. If only a small number of sites are modelled incorrectly (or are damaged) then only a small number of the matrix elements will be changed. Generally, because of the minimum norm optimisation in the updating method, all the matrix elements would be changed a little, rather than a small number of elements changed substantially. Thus the effect of any damage present would be spread over all the degrees of freedom making location difficult. Kaouk and Zimmerman  tried to overcome this problem by ensuring that the change in the stiffness matrix was low rank. This does not ensure that the change in stiffness will be local, as the stiffness change could be global but low rank. Also the method, as proposed, requires the rank of the stiffness change to be equal to the number of modes measured. Thus the localisation of damage often becomes worst when more measured modes are included; not a very satisfactory situation.
It should be apparent by now that methods that update whole mass and stiffness matrices have significant disadvantages, which are not outweighed by the major advantage of not requiring parametric models of the damage mechanisms. It is very unlikely that these methods will prove useful in the majority of structural health monitoring situations.
Parametric Methods from Model Updating and Regularisation
Most common methods in model updating rely on a parametric model of the structure [2, 3]. The measured data may be in the form of frequency response function data, or in terms of the modal data. The updating exercise minimises residuals based on the modal quantities, or the input (equation) error or output error in the frequency domain.
In model updating, the number of parameters may be reduced by only including those parameters that are likely to be in error. Thus if a frame structure is updated, the model of the beams are likely to be accurate but the joints are more difficult to model. It would therefore be sensible to concentrate the uncertain parameters to those associated with the joints. Also, damage might be more likely in the joints. Even so, a large number of potential parameters may be generated, the measurements may still be reproduced and the parameters are unlikely to be identified uniquely. In this situation all the parameters are changed, and regularisation must be applied to generate a unique solution. Regularisation generally applies extra constraints to the parameter estimation problem to ensure a unique solution. Applying the standard Moore-Penrose pseudo inverse is a type of regularisation where the parameter vector with the minimum norm is chosen. The parameter changes may be weighted separately to give a weighted least squares problem, where the penalty function is a weighted sum of squares of the measurement errors and the parameter changes. Such weighting may also be extended to include minimising the difference between equivalent parameters that are nominally equal in different substructures such as joints.
Although using parametric models can reduce the number of parameters considerably, for damage location there will still be a large number of parameters. Most regularisation techniques rely on minimum norm type solutions that will tend to spread the identified damage over a large number of parameters.
The major differences between model updating and damage/error location is that in damage location only a limited number of parameters are likely to be in error. If only these parameters are chosen then the updating would be over-determined. Unfortunately we do not know which parameters might be in error and this must be determined. Indeed determining which parameters are in error may be thought of as a form of regularisation known as subset selection .
In damage location statistical methods and performance measures have been used that work on a similar principle . Only a limited number of sites are assumed to be damaged, and the model updated based on the reduced number of parameters. This process is repeated for all possible combinations of damage site, and possibly even damage mechanism. The results from all the updated models are compared and the one that best matches the measured data is chosen.
The major problem with both subset selection and the statistical type approach, is that many smaller model updating exercises have to be performed. To optimally derive the best set of parameters, or the best damage location, requires the evaluation of many subsets of parameters. With a large number of parameters looking at all subsets of 2 or 3 parameters can become daunting. For example, if we have 100 potential parameters, then there are 100 subsets of 1 parameter, 4950 subsets of 2 parameters and 161,700 subsets of 3 parameters. Thus sub-optimal methods must be used to derive good, but not necessarily the best, subsets of the parameters.
Neural Networks and Genetic Algorithms
Neural networks and genetic algorithms have been viewed as potential saviours for the solution of the difficult problems in damage location. Although these methods may be useful in some circumstances they do not deal with the root cause of the problem. Genetic algorithms have some advantage in finding a global minimum in very difficult optimisation problems, particular where there are many local minima as is often the case in damage location. That said, the method still requires that the dynamics of the structure changes sufficiently and predictably enough for the optimisation to be meaningful. The crucial decision and difficulty is what to optimise, not the optimisation method used.
Neural networks are able to treat damage mechanisms implicitly, so that it is not necessary to model the structure in so much detail. The method can also deal with non-linear damage mechanisms easily. Models are still required to provide the training cases for the networks, and this is their major problem. There will always be systematic errors between the model used for training and the actual structure. For success, neural networks require that the essential features in the damaged structure were represented in the training data. The robustness of networks to these errors has not been tested sufficiently. Of course, the other major problem with both genetic algorithms and neural networks is that they require a huge amount of computation for structures of practical complexity, although these methods are well suited to parallel computation.
Non-Linear and Other Methods
Thus far the discussion has centred on using a linear model and linear techniques for damage location. Many damage mechanisms, such as cracks, will produce non-linear effects. Assuming the rest of the structure is linear, then there is a very local non-linearity in a predominantly linear structure. Thus it seems likely that the small changes due to damage may be more identifiable if the non-linear effects can be separated from the linear effects. The major difficulty is the lack of a standardised framework for non-linear analysis, compared with the modal and frequency domain analysis for linear systems. Methods using wavelets and other time-frequency domain transformations show promise. Methods that utilise the beating phenomena may also be used to highlight small differences in similar signals. Damage may affect the damping in a structure more than the stiffness, although methods based on this phenomena will always have the problem that the accurate, consistent measurement of damping is very difficult.
PROBLEMS AND ERRORS IN DAMAGE IDENTIFICATION
The methods outlined above have indicated some of the problems with damage identification. There are always errors in the measured data and the numerical model that affect all the algorithms. These errors, and the adequacy of the data, are now discussed.
Systematic Errors in Damage Identification
One of the major problems in damage location is the reliance on the finite element model. This model is also an important strength because the very incomplete set of measured data requires extra information from the model to be able to identify damage location. There will undoubtedly be errors even in the model of the undamaged structure. Thus if the measurements on the damaged structure are used to identify damage locations, the methods will have great difficulty in distinguishing between the actual damage sites and the location of errors in the original model. If suitable parameters are not included to allow for the undamaged model errors then the result will be a systematic error between the model and the data. Identification schemes generally have considerable difficulty with systematic errors. It is very likely that the original errors in the model will produce frequency changes that are far greater than those produced by the damage. There are two basic approaches to reducing this problem, although both rely on having measured data from an undamaged structure. The first is to update the finite element model of the undamaged structure to produce a reliable model [2, 3]. Obviously the quality of the damage location assessment is critically dependent upon the updated model being physically meaningful. Generally, this requires model validation using a control set of data not used for the updating. The second alternative uses differences between the damaged and undamaged response data in the damage location algorithm. To first order, any error in the undamaged model of the structure that is also present in the damaged structure will be removed. This does rely on the structure not changing, except for the damage, between the two sets of measurements.
The Effect of Frequency Range
The range of frequencies employed in damage location has a great influence on the resolution of the results and also the physical range of application. The great advantage in using low frequency vibration measurements is that the low frequency modes are generally global and so the vibration sensors may be mounted remotely from the damage site. Equally fewer sensors may be used. The problem with low frequency modes are that the spatial wavelengths of the modes are large, and typically are far larger than the extent of the damage. The spatial resolution of the damage identification scheme requires that there is a significant change in response between two adjacent potential damage sites. If low frequency modes are used then this resolution is closely related to the spatial wavelengths of the modes. Ultrasonics excites high frequency, very local modes which are able to accurately locate damage, but only very close to the sensor and actuator position.
Non-Stationary Systems and Damage Detection
One very difficult aspect of damage assessment is the change in the measured data due to environmental effects. This is one undesirable non-stationary effect and makes damage location difficult. Of course progressive damage is also a non-stationary phenomena, but if other non-stationary effects dominant the damage then obviously the latter will be difficult to identify. Typical of environmental effects are those in highway bridges. These bridges have been the subject of many studies in damage location, but in the UK, where most bridges are constructed using concrete, such identification has considerable problems with changes due to environmental factors . For example, concrete absorbs considerable moisture during damp weather, which considerable increases the mass of the bridge. Temperature changes the stiffness properties of the road surface, known as the 'black-top', significantly. On a hot summer's day in the UK, the road surface will provide little stiffness, but on a cold winter's day the stiffness contribution is considerable. The difficulty is trying to predict the effects of temperature and moisture absorption from readily available measurements. There are further difficulties with highway bridges because they are highly damped with low natural frequencies. They are in a noisy environment and are difficult to excite. The frequency resolution in the measurements is invariably quite low, leading to difficulty in picking up small frequency changes due to damage.
Strength vs. Stiffness
The philosophy of damage detection using measured vibration data is based on the premise that the damage will change the stiffness of the structure. In some instances there is a significant difference between strength and stiffness. Indeed, estimating the remaining useful life of a component based on conclusions from a dynamic analysis is very difficult. For example, a concrete highway bridge will have steel reinforcement cables running in channels in the concrete. The cables are tensioned, either before or after the concrete has set, to ensure that the concrete remains in compression. One major failure mechanism is by the corrosion of these cables. Once the cables have failed the concrete has no strength in tension and so the bridge is liable to collapse. Unfortunately the stiffness of the bridge is mainly due to the concrete, and so the progressive corrosion of the cables is very difficult to identify from stiffness changes. Essentially the dynamics of the bridge do not change until it collapses.
THE ROLE OF SIMULATION AND PHYSICAL TESTING
Many of the algorithms suggested for damage location are tested on simulated data. It is necessary to fully test any method on both simulated and real data. The simulated tests are able to fully exercise the location methods, with the benefit that the answer is known. In simulation, far more damage cases may be used and the effect of errors may be fully investigated. The need for real testing arises because experimental work always produces errors and problems that are unexpected. For simulation to be useful, the errors that might be expected in real structures must be simulated. Thus, adding random noise to a model of the structure and then using the same model to identify the damage in not enough! Most identification schemes are able to cope very well with random noise, and although such simulations are important parts of the overall performance assessment of an algorithm, they are not sufficient. It is vital that systematic type errors are included in the simulation. Thus, discretisation errors may be included by generating the simulated measurements using a fine finite element model; the damage mechanism introduced to generate the measurements may be different to those modelled for the identification; or boundary conditions on the structure could be changed between the measured data set and the identification.
THE WAY FORWARD
It seems that most methods have considerable difficulty. Much of this is related to systematic errors between the model and the structure and the non-stationarity of the structure. Obviously these models will improve, and model updating can help, but in some structures, for example road bridges, it is difficult to imagine that the models will be able to account for all the environmental effects. The most promising methods seem to be based on modal data and rely on the forward type identification methods. Even so, relying on low frequency vibration data will always be able to locate damage with a limited accuracy because of the global nature of the modes. One interesting area is in the interface between modal testing and ultrasonics. At some point between the two extremes there must a comprise position with reasonable localisation but with a limited set of fixed actuators and sensors. The problem in using wave propagation type methods in global applications is the large number of reflections and transmissions from components that are difficult to model at the higher frequencies.
The structural health assessment of smart structures is one application where the large number of sensors and actuators produces a huge potential benefit, but also a severe difficulty in data interpretation and processing. Algorithms should be tailored to the application and it is unlikely that a single 'best' method will ever emerge. Non-linear methods have significant potential, but the techniques are just starting to be developed. Health assessment using measured ambient excitation is even more difficult, and ways to increase the information content of the measurements would be valuable. For example, strips could be laid across a road bridge so that passing cars excite the structure in a more predictable way. The optimisation of the location of these 'exciters' would make an interesting research topic.
A large number of researchers have turned their attention to damage detection and location recently. Many of the resulting papers have demonstrated some success, but usually the examples are either simulation, very controlled experiments, or the researchers had some idea of what to expect. Robust identification techniques that are able to locate damage based on realistic measured data sets still seem a long way from reality. Certainly if our horizons are reduced, significant prior knowledge of the structure is included and sufficient measurements are taken, some progress can be made in some applications. One scenario is that damage location using low frequency vibration is undertaken to identify those areas where more detailed local inspection should be concentrated.
Dr. Friswell acknowledges the support of the Engineering and Physical Sciences Research Council through the award of an Advanced Fellowship.
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