Changes In Mode Shapes Research Articles

Effect of concentrated masses with rotary inertia on vibrations of rectangular plates

The effect of attached masses on free vibrations of rectangular plates is studied by considering rotary inertia of concentrated masses and geometric imperfections of the plate. Experiments are performed to understand the changes in natural frequencies and mode shapes. In order to validate and better understand the experiments, numerical results are also presented. Boundary conditions at the plate edges are those of simply supported plate with additional rotational springs at the plate edges. By tuning the stiffness of these springs, any boundary conditions comprised between simply supported and clamped edges can be simulated. Numerical results are successfully compared to experiments performed on a stainless-steel plate with an attached mass of different translational and rotary inertia. In particular, it is shown that a small mass placed on the diagonal of a square thin plate is enough to transform the shape of modes with one nodal line parallel to the edge, into diagonal modes, i.e. modes with diagonal nodal lines. Rotary inertia of concentrated masses reduces significantly natural frequencies. Moreover, additional modes due to the presence of large rotary inertia of concentrated masses arise, which do not appear if rotary inertia is neglected. Finally it is shown that masses placed only on one side of the plate or symmetrically (with respect to the mid-plane) placed on both sides give similar results.

A weighted balance evidence theory for structural multiple damage localization

In order to identify structural multiple damage locations, an improved evidence theory is proposed in this paper. First, the changes of frequencies and mode shapes are regarded as two different information sources, and local decisions can be obtained by using the multiple damage location assurance criterion (MDLAC) method and the frequency change damage detection method (FCDDM), respectively. Then, the local decisions are sent to a fusion center. In the fusion center, the evidence fusion technique is applied to integrate those local decisions and acquire a global decision. It is considered that the multiple evidences from different sources of different importance or reliability are not equally important when they are combined according to evidence theory, which is seldom considered in the existent combination methods. A new approach, i.e. weighted balance evidence theory (WBET), is presented to solve this problem. In order to compare the identification results under different weight coefficients, a multi-damage identification index is proposed. The simulation results demonstrate that the excellent performance of the proposed method to identify multiple damage sites as compared with the MDLAC method and the FCDDM method. In addition, analysis indicates that basic evidence combination is a particular case of the WBET method under certain condition.

Damage Identification Using Modal Data: Experiences on a Prestressed Concrete Bridge

Large scale tests with progressive damage on a prestressed concrete highway bridge have been performed to investigate the sensitivity of several damage detection, localization, and quantification methods based on modal parameters. To investigate the quality of modal parameters, the data set of one damage step was analyzed by several output-only identification techniques. Although the bridge was severely cracked, natural frequencies as well as mode shapes display only minor changes. However, the relative changes of mode shapes are larger than those observed for natural frequencies. A novel damage indicator, called mode shape area index, based on changes of mode shapes, has been developed and found as the most sensitive damage detection approach. Damage detection or localization via changes of the flexibility matrix performed better than natural frequencies or mode shapes alone. The application of the direct stiffness calculation and a sensitivity-based model update technique showed results having a high level of ambiguity about the location and quantification of damage also at the highest damage level. Evaluating the information collected in this study the test results indicate that an early stage damage identification in prestressed concrete bridges is hardly possible because of the nearly complete recovery of stiffness after closing of cracks in prestressed concrete and the effect of environmental parameters on modal data.

Considerations regarding superharmonic vibrations of a cracked beam and the variation in damping caused by the presence of the crack

A closing crack causes the dynamic behaviour of a vibrating system to be significantly nonlinear.The main distinctive features of such a vibrating system are the appearance of sub- and superharmonic resonances, and the significant nonlinearity of the vibration responses at sub- and superharmonic resonances (displacement, acceleration, strain, etc.). The nonlinear effects are much more sensitive to the presence of a crack than are either the change of natural frequencies and mode shapes or the generation of high harmonics in the spectrum of vibration at principal resonance or far from resonance. Thus, the appearance of sub- and superharmonic resonances may prove to be useful, highly sensitive indicators of a crack's presence at very early stages of its nucleation; moreover, the level of response nonlinearity in this regime may provide a quantitative evaluation of damage parameters (type, size and location). At the same time, the manifestation of nonlinear effects depends not only on the crack parameters but also on the level of damping in a vibrating system. Recent experimental tests have revealed that crack nucleation and growth result in an increase of damping in a vibrating system. Consequently, the influence of crack's parameters upon the nonlinear effects should be determined while taking into account the change of damping in a vibrating system rather than assuming either constant damping or the total absence of damping. The empirical relationship between energy dissipated in a crack and nominal stress intensity factor range, estimated by curve-fitting experimental data, has been introduced into an FE model of a beam with a closing crack, and the influence of the damping level on the nonlinear dynamic behaviour of the beam was investigated.

Damage detection in bridges using neural networks for pattern recognition of vibration signatures

Damage detection by measurement of vibration signatures is highly attractive for monitoring bridges because it provides the possibility of electronic recording combined with digital processing and report generation. However, changes in frequency and mode shape caused by damage are usually small and methods that have demonstrated successful detection have generally been confined to small-scale laboratory models. In this investigation a damage detection procedure, using pattern recognition of the vibration signature, was assessed using a finite element model of a real structure — a suspension bridge more than 100 years old. Realistic damage scenarios were simulated and the response under moving traffic was evaluated. Feature vectors generated from the response spectra were presented to two unsupervised neural networks for examination. It is shown that the sensitivity of the neural networks may be adjusted so that a satisfactory rate of damage detection may be achieved even in the presence of noisy signals.

Modeling and analysis of a cracked composite cantilever beam vibrating in coupled bending and torsion

The coupled bending and torsional vibration of a fiber-reinforced composite cantilever with an edge surface crack is investigated. The model is based on linear fracture mechanics, the Castigliano theorem and classical lamination theory. The crack is modeled with a local flexibility matrix such that the cantilever beam is replaced with two intact beams with the crack as the additional boundary condition. The coupling of bending and torsion can result from either the material properties or the surface crack. For the unidirectional fiber-reinforced composite, analysis indicates that changes in natural frequencies and the corresponding mode shapes depend on not only the crack location and ratio, but also the material properties (fiber orientation, fiber volume fraction). The frequency spectrum along with changes in mode shapes may help detect a possible surface crack (location and magnitude) of the composite structure, such as a high aspect ratio aircraft wing. The coupling of bending and torsion due to a surface crack may serve as a damage prognosis tool of a composite wing that is initially designed with bending and torsion decoupled by noting the effect of the crack on the flutter speed of the aircraft.

Direct Approach in Inverse Problems for Dynamic Systems

A direct and effective approach is presented for the inverse problem of dynamic structural systems, which is related to structural optimization, system identification, and damage detection. The structural modifications are sought for the characteristic changes assigned from the design goals or modal measurements. A finite element method is used for the system analysis and inverse problem. Mathematical programming techniques are applied for the minimization of the deviation of the finite element model from the desired inverse system, along with an objective function of least structural change. The modal method is based on the perturbation equations of a set of selected degrees of freedom and the energy equation associated with the frequency change. The mode shape change is expressed as the sum of the baseline mode shape and complementary vector, which plays a very important role in the search for the inverse solution. The linear perturbation equation is employed to get an initial approximation, which can be improved through iterations with the nonlinear perturbation equation. The proposed method does not involve the system reduction. Therefore, it is believed to be superior to conventional methods, which suffer from the residual error due to transformation of the eigenproblem.

A SIMPLIFIED NONLINEAR ANALYSIS PROCEDURE FOR SINGLE-STORY ASYMMETRIC BUILDINGS

A simplified nonlinear analysis procedure to predict earthquake responses of single-story asymmetric buildings is presented and its applicability is discussed. In this procedure, their responses are predicted through a nonlinear static analysis of MDOF model and a nonlinear dynamic analysis of equivalent SDOF model, considering the change in the first mode shape at each nonlinear stage and the effect of the first and second mode contribution. The results show that the responses of torsionally stiff buildings can be satisfactorily predicted by the proposed procedure.

Damage identification in continuous beams and frame structures using the Residual Error Method in the Movement Equation

In general, the structures are submitted during their useful life to deterioration processes that, depending on the intensity, may affect their performance and load capacity and consequently their safety. In this case, it is necessary to accomplish an inspection in order to evaluate the conditions of the structure and to locate and quantify the intensity of the damage. In this study, a method to identify and to quantify damage in structures, called Residual Error Method in the Movement Equation [Localização e quantificação de danos em estruturas por meio de suas caracterı́sticas dinâmicas. Universidade de Brası́lia, Brası́lia, DF, 2000] is evaluated, by a numerical analysis, to verify its efficiency when applied to continuous beams and frame structures. This method is based on the alteration, produced by damage, in the dynamic properties of the structures. The location of the damage is done observing the error in the movement equation of the intact structures. The structures are discretized in finite elements and the damage is introduced by a stiffness and area reduction of the elements’ cross-sections. Other two methods of damage detection used in this paper are: the Damage Detection from Changes in Curvature Mode Shapes [J. Sound Vibration 145 (1991) 321] and the indexes MAC [Proc. 1st Int. Modal Anal. Conf. 1 (1982) 110] and COMAC [Proc. 6th Int. Modal Anal. Conf. 1 (1988) 690]. Observing the obtained results, the Residual Error Method in the Movement Equation is efficient in the damage location and quantification of the studied structures.

Detection of internal delamination in multi-layer composites using wavelet packets combined with modal parameter analysis

This paper presents a study on active detection of delamination for multi-layer composites using a combination of modal analysis and wavelet transform. The analysis of modal parameters for multi-layer composites with internal delamination is carried out, and energy spectrum of wavelet packet decomposition of structural dynamic response is investigated. For several samples with different delamination dimensions and support conditions, the finite element method is used to study the relationship between delamination dimension and damage-induced change of structural physical property. Results show that delamination leads to a variation of energy dissipation in plate vibration, and it is mode-dependent. For relatively small delamination, the damage-induced changes of natural frequencies and mode shapes are too slight to be detected practically. However, by means of analysis on energy spectrum of wavelet packet decomposition even smaller delamination can be detected using the measured dynamic response signals.

DAMAGE DETECTION IN STRUCTURES: FROM MODE SHAPE TO FREQUENCY RESPONSE FUNCTION METHODS

The present paper addresses the subject of damage detection in structures. A series of numerical simulations on a simple beam are made in order to compare various damage detection methods based on mode shape changes. A generalisation of these methods to the whole frequency ranges of measurement is proposed, i.e. methods based on mode shape changes become based on operational mode shapes. The objective of such a study is to ascertain the possibility of using various damage detection methods without the need for modal identification. In recent years, the authors have developed some simple methods and tools based upon the use of frequency response functions (FRFs), which seem promising and have given good results in some practical applications. Here, some of that work is applied to the new proposed FRF-based methods which are compared and discussed. Numerical simulated examples will be used, as well as an experimental test case, where damage is inflicted in a free–free beam.

Capturing mode-switching in postbuckling composite panels using a modified explicit procedure

A postbuckling blade-stiffened composite panel was loaded in uniaxial compression, until failure. During loading beyond initial buckling, this panel was observed to undergo a secondary instability characterised by a dynamic mode shape change. These abrupt changes cause considerable numerical difficulties using standard path-following quasi-static solution procedures in finite element analysis. Improved methods such as the arc-length-related procedures do better at traversing certain critical points along an equilibrium path but these procedures may also encounter difficulties in highly non-linear problems. This paper presents a robust, modified explicit dynamic analysis for the modelling of postbuckling structures. This method was shown to predict the mode-switch with good accuracy and is more efficient than standard explicit dynamic analysis.

Non-baseline detection of small damages from changes in strain energy mode shapes

Several methods for damage detection based on identifying changes in strain energy mode shapes (SEMS) have been recently described in the literature. Most of these methods require knowing strain energy distribution for the undamaged structure (baseline SEMS). This is especially true for detection of small damages, where changes in the SEMS cannot be observed otherwise. Usually, the mode shapes from the structure under test should be compared to the baseline mode shapes to provide sufficient data for damage detection. However, these methods do not cover damage detection on structures where baseline mode shapes cannot be readily obtained, for example, structures with preexisting damage. Conventional methods, like building a finite element (FE) model of a structure to be used as a baseline might be an expensive and time-consuming task that can be impossible for complex structures. This paper suggests a method for extraction of localized changes (damage peaks) from SEMS based on Fourier analysis of the strain energy distribution. A detailed analytical proof is given for the case of a pinned–pinned beam and a numerical proof for the free–free beam. The analytical predictions have been confirmed both by the FE model and impact testing experiments on a free–free aluminum beam, including single and multiple damage scenarios.

VIBRATION ANALYSIS OF TWISTED CANTILEVERED CONICAL COMPOSITE SHELLS

The finite element method based on the Hellinger–Reissner principle with independent strain is applied to the vibration problem of cantilevered twisted plates and cylindrical, conical laminated shells. With a small number of elements, the present assumed strain finite element method is validated by convergence tests and numerical tests, comparing with the previous published vibration results for cantilevered conical shell. Computational effort and virtual storage reduce significantly due to good convergence. This study presents the twisting angle effect on vibration characteristics of conical laminated shells. Parameter studies with varying shallowness of cylindrical and conical shells are carried out. As the curvature increases, the fundamental mode shape changes from twisting mode to bending mode. For shells with a large curvature, the fundamental frequency, which is always characterized to bending mode, is almost constant independent of twisting angle. The twisting angle affects greatly twisting frequency and mode shape.

Vibration-based Damage Detection of Structures by Genetic Algorithm

Vibration-based methods are being rapidly applied to detect structural damage. The usual approaches incorporate sensitivity analysis and the optimization algorithm to minimize the discrepancies between the measured vibration data and the analytical data. However, conventional optimization methods are gradient based and usually lead to a local minimum only. Genetic algorithms explore the region of the whole solution space and can obtain the global optimum. In this paper, a genetic algorithm with real number encoding is applied to identify the structural damage by minimizing the objective function, which directly compares the changes in the measurements before and after damage. Three different criteria are considered, namely, the frequency changes, the mode shape changes, and a combination of the two. A laboratory tested cantilever beam and a frame are used to demonstrate the proposed technique. Numerical results show that the damaged elements can be detected by genetic algorithm, even when the analytical m...

Methodology to Evaluate First Transition Foundation Stiffness for Columns on Winkler Foundation

A methodology is developed in this paper to evaluate the first transition foundation stiffness for columns subjected to end concentrated loads, on a Winkler foundation, where the buckling mode shape changes. The method is based on earlier work of Naidu and Rao to predict the fundamental frequency of beam columns, wherein it was assumed that the mode shapes of vibration, initially loaded vibration, and buckling are the same. Very good predictions for the fundamental frequency of beam columns were obtained when the assumption of the same mode shapes is satisfied. In the case of the buckling of columns on Winkler foundations, at a certain value of the foundation stiffness the buckling mode shape totally changes while the mode shapes of vibration and initially loaded vibration remain the same. This led to a very large error in the predicted fundamental frequency parameter of the beam columns. This aspect is used in the present paper to evaluate the first transition foundation modulus of columns of Winkler fou...

Fuzzy logic expert system for automated damage detection from changes in strain energy mode shapes

In recent years, researchers have developed several methods for damage detection in structures employing strain energy or curvature mode shapes. Experience shows that mode shape methods are highly sensitive to measurement noise. Such sensitivity is a direct result of the second derivative applied to the displacement mode shapes, which produces curvature mode shapes. Calculation of strain energy mode shapes includes calculation of curvature as well. Consequently, the artifact false peaks (normally an indicator of damage) appear on the strain energy mode shapes, reducing accuracy of damage recognition. Combined with the intrinsic variance in damage peak shapes and amplitudes dependent upon magnitude and location of damage, the false peaks create a major problem for automated detection of damage location. Most often, a highly qualified expert is required to analyze the data and to identify and locate damage. In an attempt to replace the human expert with a computer, this paper describes a fuzzy logic expert system designed to mimic the human decision process. An expert system allows for easy encoding of expert knowledge as a set of rules. "Fuzziness" of the expert system allows better treatment of the uncertainties of the problem and to simplify the expert system itself. The fuzzy expert system has been designed based on a finite element (FE) model of a simple beam and has provided reliable detection of damage for every tested damage scenario (100% recognition). The same system recognized damages with 100% accuracy, and no false positives or negatives on mode shapes acquired by impact testing and/or non-contact laser vibrometer techniques.

Structural Damage Identification using Modal Data. I: Simulation Verification

The change of modal characteristics directly provides an indication of structural damage. Based on changes in frequencies and mode shapes of vibration, a damage identification technique is proposed in this paper for predicting damage location and severity. The method is applied at an element level with a conventional finite-element model. The element damage equations have been established through the eigenvalue equations that characterize the dynamic behavior. Several solution techniques are discussed and compared. The influence of simulated noise in the modal data is also presented. The method has been verified by a number of damage scenarios for simulated beams and has found the exact location and severity of damage. It is demonstrated that multiplying the damaged eigenvalue equations with the undamaged or damaged mode shapes provides more equations and guarantees the damage localization. The resulting equations, however, become more sensitive to the deviation of modal data and the direct solution often...

The behaviour of damage tolerant hat-stiffened composite panels loaded in uniaxial compression

Damage tolerant hat-stiffened thin-skinned composite panels with and without a centrally located circular cutout, under uniaxial compression loading, were investigated experimentally and analytically. These panels incorporated a highly postbuckling design characterised by two integral stiffeners separated by a large skin bay with a high width to skin-thickness ratio. In both configurations, the skin initially buckled into three half-wavelengths and underwent two mode-shape changes; the first a gradual mode change characterised by a central deformation with double curvature and the second a dynamic snap to five half-wavelengths. The use of standard path-following non-linear finite element analysis did not consistently capture the dynamic mode change and an approximate solution for the prediction of mode-changes using a Marguerre-type Rayleigh-Ritz energy method is presented. Shortcomings with both methods of analysis are discussed and improvements suggested. The panels failed catastrophically and their strength was limited by the local buckling strength of the hat stiffeners.

Post-Buckling of Composite I-Sections. Part 1: Theory

Atheoretical investigation of initial local buckling and post-buckling behavior of composite I-sections is presented. The equilibrium equation for initial buckling is solved both exactly and with an approximate method, namely, Galerkin’s method. In Galerkin’s method, the out-of-plane deflection of each plate element is approximated by a weighted sum of polynomial functions. The post-buckling response is studied as an extension of the approximate analysis with Galerkin’s method where now both equilibrium and compatibility equations must be solved. The bending deflection in the post-buckling regime is assumed to be a magnification of the deflection function used in initial buckling analysis. No mode-shape change is thus allowed in the post-buckling region. A polynomial type of function is also adopted for stress distribution in order to take into account the deviation from uniform in-plane load distribution in the plate elements following the onset of local buckling. The paper provides an efficient and accurate method for predicting the post-buckling behavior of composite structural sections composed of plate elements. Galerkin’s methodwas previously applied to isotropic flat plates only. The present approach is tested against a commercial code, STAGS, with a very good agreement in results and a very large saving in computer time for post-buckling analysis of an I-section.