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The paper presents the natural frequencies shift of a Warren truss with seven elements in the case that members of the structure are weakened by considering that the modulus of elasticity with a lower value. Taking into consideration that the slenderness ratio has a constant value and the transverse section of the truss bars is changed, the relative frequency shift for the first seven vibration mode were calculated by using the values obtained from numerical modal analysis. The conclusions reveal the fact from the dynamic behavior of the structure it can be obtained templates to identify the weakened truss bar of the structure.

Real beams have non-ideal boundary conditions and it is necessary to use new models to determine the real modal parameters. Models that use ideal conditions do not fully reflect reality and can lead to unsatisfactory description of the dynamic behavior. The hinged – hinged boundary conditions, which is in the focus of the paper, are not analyzed as a single beam, but as a continuous beam with three spans, free at the ends. The continuous beam with three spans is analyzed for cases in which the intermediate supports can occupy any position along the length of the beam, by an analytical solution of the problem, with the example of cases when the intermediate supports are located very close at the free ends of the continuous beam, thus simulating the real case for an hinged beam at both ends; the situation in which the intermediate supports are very close to one of the ends of the beam, thus simulating the real case of the clamped beam, with an imperfect clamped end; and the situation in which the intermediate supports are very close located anywhere on the beam length, thus simulating the hypothetic case with a continuous beam free at the ends and fix on the hinged supports. The analytic results are compared with numerical results by using finite elements method.

Continuous beams simply supported with several intermediate supports are very common in engineering achievements everywhere. The paper shows the evolution of the dimensionless wave number in 3D format, respectively of the eigenfrequencies for a continuous beam with three openings when the intermediate supports take any position inside the beam. The frequency equation for calculating the dimensionless wave number is presented and the modal function is given with an example for the case where the eigenfrequency has the maximum value at fist vibration mode.

Trusses are everywhere; they are used in bridges, antenna towers, cranes, even in parts of the International Space Station. And for good reason, they allow us to create strong structures while using materials in very efficient and cost-effective way. Trusses it is essentially a rigid structure made up of a collection of straight members. The type of truss depends on how the horizontal and diagonal beams are arranged.

We propose in this paper a method to detect cracks in circular plates clamped at the circumference. It consists in analyzing and interpreting the frequency changes that occur due to a circular arc crack. We show that the effect of the crack depends on the strain energy loss, which actually is proportional to the curvature achieved by the affected region in the radial direction. Three types of cracks are considered herein to demonstrate this dependency, differing by their length. Modal analysis is performed involving a professional simulation software for the intact plate, and that with the generated cracks. We found a perfect fit when we compared the curvature of the plate in the radial direction obtained with the analytical relation with the frequency values for the crack displaced along the radius. Therefore, using the curvature of several vibration modes we can obtain damage patterns that can be used to locate circular cracks.

In this paper, we present a simple predictive model to estimate the natural frequencies of steel cantilever beams with branched cracks. The model considers the cracked beam as a beam with constant cross-section, on which acts an equivalent bending moment. This bending moment takes into account the modification of the rigidity on the beam segment on which the crack extends in the longitudinal direction and the additional rotation that occurs in the slices located at the ends of the damaged segment. Along with the theoretical background and the deduced mathematical model, we present several cases of deterioration defined by the depth of the transverse crack branch respectively the position and the extension of the longitudinal branch. These examples prove the accuracy of the predictions regarding the change of the natural frequencies due to the damage obtained with the proposed model.

The paper present a novel non-destructive evaluation method designed to assess damage in structural elements subjected to important axial loads, as columns are. In the prior research a reliable damage detection method was developed for beams subjected to own mass, that consider the relation existing between the energy stored in the beam in certain vibration modes and the related natural frequencies. First we found the mathematical relations expressing the healthy pillar mode shapes and frequencies with respect to the top mass. Afterward, by means of FEM, we derived the natural frequencies for the numerous damage cases, in order to define the frequency shift curves. Analogous to the case of beams, the damage location is characterized by patterns that are derived from the mode shape curvatures square of the healthy beam. The damage location becomes an inverse problem while the damage position is found by interpreting frequency measurements made on the healthy and damaged beam.

Transverse cracks affect the stiffness and consequently the dynamic behavior of beams, by altering the natural frequencies. Actual cracked beam models are not able to explain in-deep the vibration mechanism and, therefore, no mathematical relation able to predict frequency changes due to damage exist. This paper proposes a new damage model which better explain the dynamic behavior of beams with open and closed cracks and a mathematical relation able to predict the frequency changes due to damage. In contrast to actual models, a global approach is used, by considering the influence of the stored energy distribution for each transversal vibration mode. Since the energy loss globally quantifies the state change due to damage, it was possible to replace the cracked beam with an equivalent one having constant but reduced stiffness. Based on it, a relation indicating the frequency drop damage was contrived and tested, by means of the finite element analysis, for various damage types and locations along the beam. Simulations have shown that accurate prediction about the natural frequency shifts can be made with the proposed relation.

This paper introduces a method to detect damages in L-shaped frames. The problem of identifying cracks in such structures arises from the low frequency changes due to damage. In the early stage, cracks are hard to be assessed. We present two distinct parts: the assessment of the dynamic behavior of L-shaped and description of the damage evaluation method. In the first part, the mathematical relations contrived for the natural frequencies and mode shapes respectively curvatures are introduced. Afterward, the distribution of the modal energy is found and associated with effects produced by a crack with a given depth in different locations. Findings are used to identify the crack location and severity. Accurate frequency evaluation, a crucial aspect since low frequency changes are expected, was made by employing an algorithm developed by the authors. The damage assessment method with the integrated frequency evaluation algorithm is employed to evaluate the health of an L-framed structure.

In this paper, a predictive model that describes the structural behavior in the absence and presence of damage, and anticipates the modal parameter changes, is introduced for weld structures. Degradation of engineering structures in the welds or HAZ due to damage affects their safety by reducing the stiffeners of structural components. In this paper was considered the kinetic energy distribution which reflects the mass participation, and the modal strain energy distribution which take into consideration the stiffeners reduction. The relation is successfully proved against FEM simulation results.

This paper present a procedure to detect wear or corrosion in structural elements based on frequency changes. Several methods to damage assessment are known, but these are mainly addressed to open cracks without considering the loss of mass. Since the mass decrease is a phenomenon associated with numerous corrosion mechanisms, the idea of this paper was to establish the influence of mass changes upon the natural frequencies of beam and to use it as a baseline in damage detection methods. Consequently, the research was conducted in two steps; first we analyzed the beam dynamics for a segment with reduced volumetric mass density but maintaining the global beam rigidity, succeeding to find the correlation between the frequency changes in the transversal vibration modes and mass loss in form of a mathematical relation. Afterwards, this relation was incorporated in the relation who predicts the frequency decreases due to stiffness diminish, in order to fit it for wear and corrosion assessment. The method’s robustness and reliability was proved by FEM simulations as well as by experiments.

This paper presents a novel non-destructive method to locate and size damages in frame structures, performed by examining and interpreting changes in measured vibration response. The method bases on a relation, prior contrived by the authors, between the strain energy distribution in the structure for the transversal vibration modes and the modal changes (in terms of natural frequencies) due to damage. Using this relation a damage location indicator DLI was derived, which permits to locate cracks in spatial structures. In this paper an L-frame is considered for proving the applicability of this method. First the mathematical expressions for the modes shapes and their derivatives were determined and simulation result compared with that obtained by finite element analysis. Afterwards patterns characterizing damage locations were derived and compared with measurement results on the real structure; the DLI permitted accurate localization of any crack placed in the two structural elements.

This paper deals with methods of interpreting the results of vibration measurement to identify structural changes in beam-like structures. We briefly presented an own developed damage assess method, that consider a large number of frequencies for the weak-axis banding vibration modes; it allows first a precise localization and afterwards evaluation of the damages. For the first step, recognition of the damage position, we introduce an algorithm implemented in C++ with the interface done using EXCEL features, indicating by one number the damage probable position, based on the Minkowski metrics. To avoid uncertainties, a graphic representation of all results is also presented. The method is tested for values determined by calculus for a randomly selected location, with and without measurement results debased by noise, proving its reliability.

Corrosion as material destruction affects the safety of structures, leading to more serious consequences than the simple loss of a mass. The effect of corrosion on the dynamic behaviour of structures act in two ways: the loss of mass increases the natural frequencies, opposite to the effect of stiffness loss. This paper present the results of researches made to extend the mathematical relation presenting the influence of damage location and severity on the natural frequency changes, by adding the effect mass loss. Therefore we modeled the beam once with the discontinuity and loss of mass, afterwards the damaged segment is replaced by an intact one but having the mass similar to that of the damaged segment. This permitted to plot frequency shift curves for both cases and to contrive the relations defining that curves. Finally a relation summarizing the both effects was contrived; it was confirmed both by numerical simulations and experiments.

The paper presents a method to assess the crack depth in beams for which the damage location is known. Previous researches lead us to a method to identify damage locations in beams, based on a relation providing frequency changes in respect to damage location and depth. Separating the two variables it is possible to find first the damage location, and afterwards to derive the term controlling the severity. Comparing it with values indicating the frequency shift in respect to damage depth, the severity can be assessed. The paper presents a relation reflecting this dependency for any cross-section type, involving the static deflection for the healthy and damaged beam alone; it has a general character, being not influenced by the cross-section shape.

This paper is concerned with vibration based non-destructive evaluation of structures, with a focus on quantitative assessment of damage. In previous works, a reliable method to locate open cracks in beams has been proposed and tested using both data from numerical simulations and laboratory experiments. It bases on the fact the natural frequency of a bending vibrations mode attend different changes, depending on the loss of stored energy for the slice on which the damage is located. As bigger the mode shape curvature value on that location, so bigger the loss of stored energy and consequently the natural frequency decrease in that mode. Analyzing the natural frequency changes for a larger series of vibration modes, it’s possible to precisely locate damages. The authors succeed to find a single mathematical relation describing the frequency changes for all bending vibration modes, involving one term defining damage’s location and one defining its depth. While the first term changes for different modes, being defined by the mode shape curvature, the second maintain its value for all modes, being affected just by damage depth. This finding permits decoupling the location issue with that of quantitative assessment of damage. Latest researches, presented in this paper, succeed by finding the relation between the second term of the relation and some mechanical characteristics of the beam, i.e. extending the proposed method by including evaluation of damage severity. The approach is illustrated on a cantilever beam, modeled with 3D elements.

The paper presents a research performed by the authors, in order to establish the optimal element dimensions for a given beam, so that FEM simulations should offer highest accuracy by minimum time consumption. To achieve this desire, we made FEM simulations with elements of various dimensions for seven cross-section dimensions

The paper reflects the research performed by the authors, in destined to find out the proper beam model, in concordance with its geometrical characteristics. This knowledge is needed to establish the adequate relations in the difficult process of damage detection. First, we calculated the natural frequencies for two cases: the Euler-Bernoulli model and the Share model. Afterwards, using FEM simulations we determined these frequencies again. Comparison between analytic results and FEM analysis allow seeing which theoretical models best fit for the various investigated cases.

This paper presents the natural frequencies distribution on the damaged beams. The frequencies distributions was analyzed for three types of beams: cantilever, double clamped and simple supported. The analysis was performed using the finite element method (FEM), for the undamaged beam and for the beam having damages at around 190 locations, considered one by one, and nine levels of depth. The authors deduced analytical relationships for damaged beam frequency shift and presents the results.

This paper introduces a method to identify damages in beam-like structures by analyzing the natural frequency changes of the first six transversal vibration modes. A correlation between the damage location and frequency change is established for each mode separately, by considering the modal strain energy stored in that location. The mathematical relation describing this correlation is used to characterize the dynamic behavior of a beam with a damage of known position and to derive its Damage Location Indicator (DLI) as a six-term vector. The method consists in comparing the vectors describing the damage at any possible location along the beam with the Damage Signature (DS), which is achieved from the measurements that compare the beam's frequencies in healthy and damaged state. A modified Kullback-Leibler Divergence is used to assess the damage location. In order to permit early damage assessment, an improved frequency evaluation algorithm was developed. It is based on signal truncation and consequent spectral lines rearrangement, in order to accurately find the strongest spectral components. The effectiveness of the proposed method is demonstrated by simulations and experiments.

This paper proposes a vibration-based damage detection method designed for structural elements subjected to important vertical loads, as columns or pillars. The method is based on the relation existing between the energy stored in the pillar in several vibration modes and the corresponding natural frequencies. For a certain mode, this energy results as the sum of the energies stored in all pillar slices, being dependent on the rigidity and the squared of the corresponding mode shape curvature. This means that the energy distribution along the pillar is different for each mode. Thus, reducing the rigidity of one slice due to damage, the frequencies will decrease in different ways, depending on the slice location. This fact permits to contrive patterns able to characterize the effect of damage at any location along the pillar. Since the mode shapes (and the natural frequencies) are influenced by inertial forces, but in the meantime by the compression and shear forces induced by the top mass, the patterns have to be derived for each load case. The paper presents a simple mathematical expression able to predict frequency changes if damage occurs at any location along the pillar and for any top mass value. Patterns that characterize the damage location are consequently derived by using the squared mode shape curvatures of the healthy beam. The damage location becomes an inverse problem, it being found by interpreting the results of frequency measurements for the healthy and damaged state. The process of damage location is exemplified by numerical simulations.

Degradation of engineering structures due to corrosion affects their safety by reducing the cross-section of structural components and altering the material's mechanical characteristics. These parameter changes are observable in the shift of the natural frequencies. In the study presented in this paper, it was demonstrated that the kinetic energy distribution reflects the mass participation, thus being able to predict frequency changes due to mass loss, while the modal strain energy distribution can be properly used to indicate the location of the damage. As a result, two mathematical relations were developed by the authors, predicting the frequency changes due to the main effects of corrosion: loss of mass and loss of stiffness. Degradacija tehničnih konstrukcij zaradi korozije vpliva na njihovo varnost, saj zmanjšuje presek nosilnih komponent in spreminja mehanske lastnosti materiala. Te spremembe parametrov lahko spremljamo prek premika lastnih frekvenc. Namen članka je predstavitev matematičnih relacij, ki napovedujejo spremembe frekvenc zaradi splošne ali lokalne korozije na podlagi znane izgube mase in zmanjšanja upogibne togosti. Analitična raziskava se začne z vplivom mase rezin na različnih mestih nosilca. Rezultati so pokazali, da položaj rezine določa količino energije, ki prispeva k celotni energiji, shranjeni v nosilcu v danem vibracijskem načinu. Količina energije v različnih načinih se razlikuje in je odvisna od kvadrata oblike načina na mestu rezine. Na ta način je bilo pokazano, da porazdelitev kinetične energije odraža prispevek mase v celotni shranjeni energiji in tako omogoča napovedovanje spremembe frekvenc zaradi izgubljene mase.

The paper presents a method to locate discontinuities in form of transversal cracks in beams, based on vibration measurements. Patterns characterizing frequency changes of the first ten weak-axis bending vibration modes are determined for all possible locations on the structure, using a relation contrived by the authors. It base on the correlation between the strain energy stored in a segment of the beam, which is proportional with the square of the mode shape curvature of the considered vibration mode at that location, and the frequency change for this mode by the occurrence of a discontinuity on that segment. The patters consist from a series of ten values representing the normalized relative frequency shifts for the first ten vibration modes. For a structure similarly supported, by continuous or periodical measurements, potential frequency changes can be detected. By processing these data the so-called damage location index for that crack is found out, also as a series of ten values representing the relative frequency shifts of the ten vibration modes. To precise locate the crack a pattern matching method involving the database with all possible patterns and the damage location index is used. Knowing the location, it is easy to determine by analytic calculus the crack depth. The method is easy to be used, provide accurate results, demands modest computational effort and has the advantage that the measurements may be carried out in situ with rather simple equipment. The method was validated by experiments.

The paper presents a method to detect, locate and evaluate damage severity of Euler-Bernoulli beams, based on how natural frequencies change due to damages. Previous researches that dealt with this issue focused only on quantitative changes, mainly considering a global stiffness reduction in the damaged area. The authors have contrived a correlation between the strain energy stored in a segment of the beam, which is proportional with the mode shape curvature of a considered vibration mode at that location, and the frequency change for this mode if damage appears on that segment. This reveals that for an element of the beam, the stiffness change of a certain mode for a given damage varies between zero and a maximum, depending solely on the location of that element. Moreover, one has to consider different stiffness changes for a damaged element placed on a certain location, depending on the vibration mode. This rule how frequencies of various modes change due to damage are used to create patterns, based on relative frequency shifts, which characterize damaged beams in respect to defect location and severity. The method was validated by numerous experiments, which proved its accuracy and reliability.Copyright © 2012 by ASME

The paper presents a method to assess damages in beams, based on how natural frequencies of bending vibration modes change due to damages. The authors have contrived a correlation between the strain energy stored in a segment of the beam, which is proportional with the mode shape curvature of a considered vibration mode at that location, and the frequency change for this mode if damage appears on that segment. For a certain mode, damages placed on inflection points of the mode shape curvature, where the strain energy is null, will not produce changes in frequency, while damages placed on maxima will produce the highest changes in frequency. For other locations of damage, the frequency shift will be proportional with the mode shape curvature of the vibration mode at that location. We worked out a general relation, which gives the frequency shift of all bending modes, with one coefficient depending on the support type. To evaluate damages, we determine analytically the relative frequency shift as ratio between the frequency change and the natural frequency of the undamaged beam, for the first ten vibration modes, considering various damage depths and locations. Comparison of results with that obtained by measurements on the real beam permits detection, location and assessment of damages in beams with high accuracy. The method was validated by experiments.

Damage detection methods based on vibration analysis make use of the modal parameter changes. Natural frequencies are the features that can be acquired most simply and inexpensively. But this parameter is influenced by environmental conditions, e.g. temperature and operational loads as additional masses or axial loads induced by restraint displacements. The effect of these factors is not completely known, but in the numerous actual research it is considered that they affect negatively the damage assessment process. This is justified by the small frequency changes occurring due to damage, which can be masked by the frequency shifts due to external loads. The paper intends to clarify the effect of external loads on the natural frequencies of beams and truss elements, and to show in which manner the damage detection process is affected by these loads. The finite element analysis, performed on diverse structures for a large range of temperature values, has shown that the temperature itself has a very limited effect on the frequency changes. Thus, axial forces resulted due to obstructed displacements can influence more substantially the frequency changes. These facts are demonstrated by experimental and theoretical studies. Finally, we succeed to adapt a prior contrived relation providing the frequency changes due to damage in order to fit the case of known external loads. Whereas a new baseline for damage detection was found, considering the effect of temperature and external loads, this process can be performed without other complication.