Topics in Modal Analysis, Volume 7

In the recent years, vibration-based identification techniques have attracted the attention of the civil engineering community, as these methods can be naturally incorporated into automated continuous structural health monitoring procedures. It is a generally accepted approach to model the damage and deterioration of a structural element through stiffness reduction. For this reason, a feature tailored so as to be well correlated to the expected differences between the undamaged and damaged flexibility matrices, such as the recently proposed Flexibility Proportional Coordinate Modal Assurance Criterion (FPCOMAC), is ideally suited to be exploited as damage sensitive feature. We present a statistical pattern recognition based damage detection method that employs FPCOMAC as damage sensitive feature. The proposed methodology is executed according to the training and testing phases typical of the pattern recognition framework. Particular effort is devoted to test the ability of the method to correctly identify the damage when response time histories used in the training are measured in different environmental conditions. The formulation is derived considering a shear-type structural system. Results obtained by considering a 7 DOFs shear-type system prove the efficiency of the method in detecting and locating the damage, irrespective of damage severity and environmental effects, under the conditions that the damage amount is greater than the structural variations caused by the external factors and the amount of data is reasonably large.

Robust damage detection algorithms are the first requirement for development of practical structural health monitoring systems. Typically, a damage decision is made based on time series measurements of structural responses. Data analysis involves a two-stage process, namely feature extraction and classification. While classification methods are well understood, no general framework exists for extracting optimal, or even good, features from time series measurements. Currently, successful feature design requires application experts and domain-specific knowledge. Genetic programming, a method of evolutionary computing closely related to genetic algorithms, has previously shown promise as an automatic feature selector in speech recognition and image analysis applications. Genetic programming evolves a population of candidate solutions represented as computer programs to perform a well-defined task such as classification of time series measurements. Importantly, genetic programming conducts an efficient search without specification of the size of the desired solution. This preliminary study explores the use of genetic programming as an automated feature extractor for two-class supervised learning problems related to structural health monitoring applications.

Sensitivity-based model error localization and damage detection is hindered by the relative differences in modal sensitivity magnitude among updating parameters. The method of artificial boundary conditions is shown to directly address this limitation, resulting in the increase of the number of updating parameters at which errors can be accurately localized. Using a single set of FRF data collected from a modal test, the artificial boundary conditions (ABC) method identifies experimentally the natural frequencies of a structure under test for a variety of different boundary conditions, without having to physically apply the boundary conditions, hence the term “artificial.” The parameter-specific optimal ABC sets applied to the finite element model will produce increased sensitivities in the updating parameter, yielding accurate error localization and damage detection solutions. A method is developed for identifying the parameter-specific optimal ABC sets for updating or damage detection, and is based on the QR decomposition with column pivoting. Updating solution residuals, such as magnitude error and false error location, are shown to be minimized when the updating parameter set is limited to those corresponding to the QR pivot columns. The existence of an optimal ABC set for a given updating parameter is shown to be dependent on the number of modes used, and hence the method developed provides a systematic determination of the minimum number of modes required for localization in a given updating parameter. These various concepts are demonstrated on a simple model with simulated test data.

Sensitivity-based model error localization and damage detection is hindered by the relative differences in modal sensitivity magnitude among updating parameters. The method of artificial boundary conditions is shown to directly address this limitation, resulting in the increase of the number of updating parameters at which errors can be accurately localized. Using a single set of FRF data collected from a modal test, the artificial boundary conditions (ABC) method identifies experimentally the natural frequencies of a structure under test for a variety of different boundary conditions, without having to physically apply the boundary conditions, hence the term “artificial.” The parameter-specific optimal ABC sets applied to the finite element model will produce increased sensitivities in the updating parameter, yielding accurate error localization and damage detection solutions. A method is developed for identifying the parameter-specific optimal ABC sets for updating or damage detection, and is based on the QR decomposition with column pivoting. Frequency response data collected from a simple laboratory experiment is used, along with the corresponding finite element-generated data, to demonstrate the effectiveness of the ABC-QR method.

In model updating the aim is to reach the optimal correlation between FE model and test data by modifying model parameters. The traditional solution to an updating problem is obtained by non-linear optimization processes governed by the selected updating parameters and target responses. Due to imprecision and lack of information in measurements, inaccuracy in model and several possible updating parameters a wide range of potential solutions to the same problem is present. This paper proposes a technique in which the updating problem is solved in one step using modal properties, i.e. natural frequencies and mode shapes, as target responses. The method utilizes mode shape sensitivity equations combined with the Bernal Projection equation to detect mass and stiffness discrepancies between model and experimental data. The proposed one step solution will only work accurately in cases where a reasonable FE model is available. The technique is demonstrated on simulated data of modal properties before and after mass perturbation of a glass plate. The data is polluted with noise in the range of what can be expected from real measured data.

The new transport aircraft A400M is designed for worldwide military and humanitarian missions. It is a turbo-propeller aircraft able to fly high, fast and over long distances and to take off and land on short rough runways. Regarding interior noise and vibration harshness, the A400M meets the state-of-the-art requirements. A disused A400M pre-series fuselage is currently incorporated into a vibro-acoustic test environment. Airbus has made the 30 m long, 6 m wide and 13 ton heavy fuselage part available for research at Helmut-Schmidt-University of the Federal Armed Forces in Hamburg (HSU). The possibility to analyse the interdependency of acoustics and structure in a controlled laboratory environment on an entire fuselage of the most modern propeller-driven large transport aircraft is a worldwide novelty. The enhancement of passive and active noise reduction technologies for aircraft structures will be one of the potential research key aspects. Transporting the fuselage from the Airbus production site in Bremen to the HSU campus was a logistical master piece – whether by land, air or sea. This paper gives a brief overview of a few selected past and present aircraft interior acoustics research projects, describes the transport of the A400M fuselage to HSU, highlights the first steps in the design of the test rig and introduces a potential scenario for a vibro-acoustic test series taking advantage of this unique test environment.

The vibro-acoustic behavior of large lightweight structures such as aircraft fuselage has to be tested, especially to improve active and passive means. Acoustic transmission laboratories with reverberant and anechoic rooms supply reliable test conditions for aircraft panels of a size up to several square meters. Shock mounts support these panels to realize free or pinned boundary conditions with minor damping, which is sufficient for high frequencies. For low frequencies the vibration of the whole fuselage has to be considered. Therefore the support of the panel under test should provide the same dynamic mass as the fuselage to which the panel will be connected in the aircraft. This paper describes two examples how to realize such boundary conditions. Therefore the dynamical behavior of adjacent flexible structural elements is simulated. Force and acceleration sensor signals are combined with real time control on a rapid control prototyping system. Electrodynamic shakers are used as actuators. In the first experiment the performance is demonstrated by the simulation of a variedly excited damped two mass-spring system with an adaptive LMS algorithm in time domain; in the second experiment by the test of a flexible beam under harmonic load regarding the second eigenmode with internal model in frequency domain.

Automotive manufacturers try to decrease emissions and fuel consumption by downsizing the engine. This means the number of pistons and the swept volume are reduced. Unfortunately, this increases the vibrations which are caused by the oscillating movement of the pistons. The vibrations are transferred through the power train and the car body to the passenger seats. To reduce these vibrations absorbers are installed into the power train. In this paper a vibration absorber is simulated by general purpose multi-body simulation software. Swing arms which are clamped to a disc in radial direction are modelled as flexible bodies. Each swing arm has got a mass at its end which is in form of a rigid body. Because of the centrifugal force which increases the geometric stiffness of the swing arms, the vibration absorber’s eigenfrequency continuously adapts to the current engine rotational speed. Thus the adaptive vibration absorber is able to reduce vibrations from idle up to maximum speed. The paper describes the multi-body system with flexible bodies, the linearization of the nonlinear equations of motion and the results of a parameter study.

Optimization of actuator and sensor positions is a crucial part of the design process of an active noise reduction (ANR) system. A common variant is to identify a set of actuator and sensor positions from a superset of possible locations that are specified by construction and design factors. Effective tools for this task are optimization algorithms like for example the genetic algorithm. A proven method to make the active noise control algorithm robust against disturbances and to transfer the zone of noise reduction from the error microphones to a wanted area is the implementation of weighting matrices in conjunction with a frequency domain FxLMS control algorithm. Computation of the weighting matrices is also done with the help of non-linear optimization algorithms. This paper shows a method how to combine both optimizations into one single optimization process. Furthermore, different restrictions like maximum loudspeaker actuation are taken into account. The described method is implemented for an example of an industrial working station. In this context it is compared to both a sole component optimization and a sole weighting optimization.

This paper deals with semi-active vibration reduction by means of piezo-actuators shunted to time-variant impedances. Particularly, attention is focused on the Synchronized Switch Damping on Inductance technique applied to single mode control in presence of a random disturbance. Drawbacks of the mentioned method are evidenced, showing that sometimes it can fail. Therefore, an alternative method (which actually is an evolution of SSDI) is proposed and its effectiveness is proved numerically through a model. Such a model is validated experimentally within the paper.

In this work, a fully geometrically nonlinear dynamic finite element (FE) model, which considers the kinematics of small strains but large rotations, is developed for transient analysis of piezolaminated thin-walled structures based on first-order shear deformation (FOSD) hypothesis. Linear electro-mechanically coupled constitutive equations and the assumption of linearly distributed electric potential through the thickness of the piezoelectric layers are employed. An eight-node quadrilateral plate/shell element with five mechanical degrees of freedom (DOFs) per node and one electrical DOF per smart layer is adopted in the finite element formulation. The second order differential dynamic equation is solved by the central difference algorithm. The mathematical method is validated by transient analysis of three different examples of a beam, a plate, and a cylindrical shell. The results illustrate that the geometrical nonlinearity affects the structural dynamic responses significantly.

Modal testing is routinely performed on space craft and launch vehicles to “verify” (calibrate and validate) analytical models. Large multi-component structures such as the Space Transportation System (STS) or “Space Shuttle,” and the newly proposed NASA Space Launch System (SLS), are impractical to test in their assembled configurations. Alternatively, substructure testing of these multi-component structures has been performed and used to calibrate/validate analytical models of the substructures, which are then assembled to analyze system response to applied loads. This paper describes a methodology applicable to uncertainty quantification (UQ) of reduced (modal) models of linear (or linearized) finite element models of substructures, as well as reduced (stochastic neural net (SNN)) models of nonlinear substructures, such as joints. The UQ at reduced substructure levels is propagated to higher levels of assembly by efficient means, enabling predictive accuracy assessment at the coupled system level. UQ is based entirely on comparisons on analysis and test data at the substructure level so that recourse to the specification and quantification of element-level random variables is not necessary. Examples are presented to illustrate the methodology.

Structural energy dissipation pattern is modified by the presence of discontinuities like a crack. Crack nucleation and growth reduces the structural stiffness which makes this effect useful as a damage indicator. Computational models have become the main tool for understanding the behavior of complex structures when experimental evaluation can be difficult to perform. However, many of this classical numerical analysis assumes a deterministic model and almost nothing is understood about the effect of uncertainty in the parameters, external forces and boundary conditions. This work presents a study about the energy flow patterns caused by localized damage in structures like rod, including uncertainties in a geometric parameter. The problem is solved in two steps. First, the structure is modeled by the Spectral Element Method (SEM). The mean and variance of displacement responses are obtained by using the Polynomial Chaos (PC) expansion. In PC the stochastic solutions are expanded as orthogonal polynomials of the input random parameters. Second, by using the displacements obtained in the step before, the mean and variance of energies are calculated by applying the expectation into the equations of energy density and energy flow. However, this approach produces unusual equations for expected values and covariances. Like, the expected value for a product of three random correlated variables, whose solution includes the covariance between one variable and a product of two others variables. A formulation is developed and proposed to solve this problem. Monte Carlo Simulation (MCS) is used to validate the results obtained by these solutions. Numerical examples are analyzed for some different cases, which present good approximation as compared with MCS results.

Abstract This historical work couples model order reduction, damage detection, dynamic residual/mode shape expansion, and damage extent estimation to overcome the incomplete measurements problem by using an appropriate undamaged structural model. A contribution of this work is the development of a process to estimate the full dynamic residuals using the columns of a spring connectivity matrix obtained by disassembling the structural stiffness matrix. Another contribution is the extension of an eigenvector filtering procedure to produce full-order mode shapes that more closely match the measured active partition of the mode shapes using a set of modified Ritz vectors. The full dynamic residuals and full mode shapes are used as inputs to the minimum rank perturbation theory to provide an estimate of damage location and extent. The issues associated with this process are also discussed as drivers of near-term development activities to understand and improve this approach.

Vibration health monitoring of structures involves solving inverse structural dynamic problems that are often mathematically ill-conditioned. Many solution approaches have been developed, of which dynamic residual minimization yielded some of the most computationally attractive algorithms. Minimum Rank Perturbation Theory (MRPT) offers a closed form solution of model matrix perturbation using only a subset of the measured modes equal to the rank of perturbation. The initial formulation of the theory assumed symmetric model matrices in the equations of motion, but was later extended to deal with skew-symmetric matrices. The development led to the discovery of a new matrix property known as “null-symmetry”, which helped explain how MRPT preserves symmetry and skew-symmetry in its solution. To handle incomplete measurements, an iterative algorithm based on MRPT was developed using repeated substitution of the dynamic model reduction transformation to achieve convergence. However, the algorithm can become unstable, and hence does not guarantee convergence. Driven by the limitation, dynamic least squares method was developed to solve for a robust set of parameter perturbations in a least square sense regularized by the reduced dynamic residual constraint. The algorithm is capable of estimating mass and stiffness parameters simultaneously under noise and uncertainty without repeatedly solving the eigenproblem.

The incomplete measurement problem poses a significant obstacle for both model correlation and structural health monitoring (SHM). In practice, information about the health of a structure must be ascertained using measurement data from only a limited number of sensors. Several approaches to this problem have been proposed. One approach involves a reduced order model, such as that obtained using methods such as the Guyan reduction, the Improved Reduced System Model, or the System Equivalent Reduction Process. A second approach, which this work considers, involves an expansion of the test data to match a higher-fidelity model. This work presents a damage detection method utilizing Ritz vectors and the Method of Expanded Dynamic Residuals (MEDR). Ritz vectors have several advantages over eigenvectors for application to damage detection, including lower sensitivity to noise, and, as a result of their load-dependent nature, a greater sensitivity to localized damage. The MEDR restricts identified damage locations to those where there is physical connectivity, which eliminates the “smearing” that plagues direct expansion methods, and provides a physically meaningful estimate of the damage location.

LA-UR-12-25460

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Damping forces are typically ignored during the Finite Element Analysis (FEA) of mechanical structures. In most real structures, it can be assumed that there are several damping mechanisms at work, but they may be difficult to identify, and even more difficult to model.

Since both mass & stiffness matrices are available during an FEA, a common method of modeling viscous damping is with a

proportional

damping matrix. That is, the viscous damping matrix is assumed to be a

linear combination

of the mass & stiffness matrices. Therefore, in order to model viscous damping with a

proportional

damping matrix, the two constants of proportionality must be determined.

In this paper, a least-squared-error relationship between

experimental

modal frequency & damping and the proportional damping

constants of proportionality

is developed. An example is included in which experimental modal parameters are used to calculate the constants of proportionality. The modal parameters of an FEA model with proportional damping are then compared with the original experimental modal parameters.

The elevated specific strength of fiber reinforced plastics (FRP) is the prominent drive for their ever-growing applications. Their inadequate vibrational damping properties prevent them from replacing conventional metal alloys for certain structural applications. In this study we attempt to utilize a low temperature hydrothermal synthesis to grow ZnO nanorods on the surface of woven carbon fibers and implement the resulting hybridized fibers in an epoxy matrix composite. X-ray diffraction and scanning electron microscopy are carried out to study the morphology of the surface-grown nanorods and their adhesion to the substrate carbon fibers. Two-layered hybrid composite laminas are tested for their structural damping properties using dynamic mechanical analysis (DMA). It is observed that the ZnO nanorods can enhance the damping figure of merit of the composites by 40 % without a major delineation in the storage modulus. The enhanced damping performance can be attributed to the additional surfaces manifested by the presence of the ZnO nanorods and, consequently, augmenting the sliding and frictional mechanisms. Furthermore, ZnO as piezoelectric material has the energy scavenging advantages over other 1D nanostructures which ultimately constitutes the fabricated hybrid composites as a multifunctional structural material for energy harvesting applications.

During a field test campaign, Sandia National Laboratories acquired operational data both in parked and rotating conditions on a modified MICON wind turbine with the Sensored Rotor 2 experiment. The objective of the test campaign was to acquire data to develop advanced system identification and structural health monitoring techniques. The data includes wind speed, tower deformations, low and high speed shaft rotational speed measurements as well as accelerations and strains on different locations of the blades. Applying Operational Modal Analysis on such data represents a difficult task due to the strong influence of rotor harmonics on the measured data. Accurately identifying and removing the harmonics is required to perform modal parameter identification. In this paper, data acquired with the turbine in both parked and operating conditions will be analyzed and the modal results compared. Several harmonic removal techniques will be applied on the operational data and their efficiency to solve this specific problem analyzed. In addition, a new enhanced identification technique will be applied, that improves the parameter estimation accuracy in the case of very noisy data and also provides uncertainty bounds of the parameters.

This work describes an automatic method for removing modulated sinusoidal components in signals. The method consists in using the Optimized Spectral Kurtosis for initializing Series of Extended Kalman Filters. The first section is an introduction to vibration applications with Kalman Filters and modulated sinusoids. The detection process with OSK is described in the second section. The third section concerns the tracking algorithm with SEKF for amplitude and frequency modulated sinusoidal components. The last section deals with the complete process illustrated with an experimental application on a rotating machine.

Brush cutter powered by engine has been widely used on a daily basis for mowing of road shoulders and gardens. People who regularly operate brush cutter are at risk of developing Raynaoud’s disease due to large vibration. Such vibration disorder can be prevented by reducing the handle vibration by good use of structural modification. In this study we focused on rubber bushes which support the drive shaft in pipe to transmit the engine power to the cutting blade. First the dynamic characteristics of the brush cutter such as Operational Deflection Shape are grasped and then the hardness and the placement of rubber bushes are optimized to reduce the handle vibration effectively.

In this paper, design efforts to develop a custom test setup for measuring dynamic stiffness of vibration isolators are presented. The setup is designed to conduct dynamic stiffness measurements for various static preload values and over a certain (target) frequency range. Direct Method has been selected among the methods defined by standards found in the literature. In order to investigate the effect of basic design parameters of the test setup on its overall performance, an equivalent eight degree of freedom lumped parameter model of the test setup is used which takes into account the basic dimensions and materials used for main structural components of the proposed setup design as well as the inertial characteristics of the isolators. Using the equivalent model, virtual tests are performed and the accuracy of the test setup is studied for various testing scenarios. A major work that is conducted as part of this work is to come up with a procedure that will enable tuning of the setup parameters such that the percent error on measured dynamic stiffness of various types of isolators are minimized for the case when various levels of error are present in measured displacement and force amplitudes.

Recent advances in various micro-manufacturing techniques have enabled utilization of miniature devices in numerous applications. However, testing, modeling, and predicting performance of these structures still poses various challenges. Experimental modal analysis techniques have been widely used to obtain the dynamic characteristics of structures; however, having very high natural frequencies, small vibration amplitudes, and high compliance, miniature structures require additional care during modal testing. Although recent developments in sensor technology enabled to obtain accurate high frequency vibration measurements during modal testing of miniature structures, excitation of miniature structures without any damage and within high frequency range in a reproducible manner is still being investigated. This paper presents design, development, and performance evaluation of a custom-made impact excitation system that enables repeatable, high bandwidth, single-hit impacts, and controllable impact force for modal testing of miniature structures. The system is equipped with a miniature force sensor attached to a custom designed flexure and an automated release mechanism driven by an electromagnet. The excitation bandwidth and the impact force exerted to the test structure can be controlled through control parameters: initial displacement given to the flexure and gap between the impact tip and the test surface.

Many structures experience random vibration caused by the rapid flow of air over the external surface of the structure. One example of this “aerodynamic excitation” occurs when missiles fly through the atmosphere en-route to target in powered flight, or slung to the undercarriage of an aircraft. In most cases, it is necessary to carry out laboratory testing in order to qualify the design of the structure and to assess the pedigree of the manufacturing and assembly process. The laboratory test should replicate, as closely as possible, the damage potential of the in-flight environment. The traditional method of replicating the aerodynamic induced vibration in the laboratory is to rigidly attach the structure to a large electrodynamic shaker and to subject the structure to random vibration. This testing methodology is inadequate and non-representative for two main reasons: (1) the excitation mechanisms are very different, i.e. through a distinct region when attached to a shaker compared to distributed excitation over the entire outer surface in-flight, (2) The boundary conditions are very dissimilar, i.e. attachment to a very high mechanical impedance shaker compared to “free” flight. There is much evidence to show that this testing methodology often leads to overly severe tests. Furthermore, the test program can be costly and time-consuming as tests are often carried out sequentially in three orthogonal axes. In addition, tests have to be repeated with different response control accelerometers as it is impossible to maintain in-flight responses all over the structure simultaneously due to the two reasons given previously. This paper details research carried out to replace the traditional rigid shaker approach to one with the structure “freely” supported and excited at multiple locations simultaneously using Multi Input Multi Output (MIMO) vibration control. The research focusses on analytical models in Matlab and Ansys to carry out “virtual tests” in order to demonstrate issues with the current testing methodology and to highlight the benefits of a new approach. Results from the analytical models show significant improvements in degree-of-replication and would result in faster and more cost effective laboratory testing.

The application of experimental modal analysis methods to nonlinear structures (sometimes referred to as “nonlinear modal testing” – NLMT) is not a new field, but only in the past few years has it become mature enough to be approached in a systematic way. Many methods have been developed over the years for dealing with nonlinearities in structural dynamics, but nonlinearity is an extremely complex phenomenon with so many aspects and consequences that is not possible to have a single method capable to deal with all of them. Rather than taking a holistic approach, it is perhaps useful for the engineer to have a set of mathematical tools to analyse separate subsets of the whole problem, i.e. one being within the scope of each individual investigation. The main objective of this paper is to provide a modular framework from which the engineer can choose the most appropriate method to retrieve information about an examined nonlinearity, based on the type of information needed and the available data set. This is achieved by performing a breakdown of the nonlinear modal analysis process into four main stages: detection, localisation, characterisation and quantification – each of these providing a different level of insight into the problem. A review of currently-available algorithms applicable for these four categories is presented, as well as their application to two simple case studies.

The SCARC (Simulated Carbon Ash Retention Cylinder) model is being developed to model rock stressing, and fracture development, when underground ground voids are backfilled with mine wastes. SCARC specimens are hollow tubes, cast with cementitious materials, and are filled with different blends of mine wastes, such as tailing and post-processed slurry. To monitor the strain history of the concrete cylinder, fiber optical sensors are wrapped around the tube exteriors. This distributed fiber optic layout consists of eight FBG strain gages in a single loop, permanently affixed to the SCARC cylinder. The fiber optic sensors monitor the material expansion characteristics of the initial filling stage of the waste material, and perpetually monitor the induced strain over a 24 h period. The initial experiment, of SCARC specimens, indicates that fiber optic sensors can successfully monitor, and accurately report, the strain history of the cylinder, as well as detect the fracture location within the cylinder, by the distributed sensor array. The results indicate the potential of fiber optics as embedded sensors for monitoring the backfilling of waste into abandoned mines, which is suggested as a mine stabilization technology.

Among numerous damage identification techniques, those which are used for online data-driven damage identification have received considerable attention recently. One of the most widely-used vibration-based time-domain techniques for nonlinear system identification is the extended Kalman filter, which exhibits a good performance when the parameter to be identified is a constant parameter. However, it is not as successful in identification of changes in time-varying system parameters, which is essential for real-time identification. Alternatively, the extended Kalman-Bucy filter has been recently proposed due to its enhanced capabilities in parameter estimation compared with extended Kalman filter. On the other hand, when applied as the excitation in some attractor-based damage identification techniques, chaotic and hyperchaotic dynamics produce better outcomes than does common stochastic white noise. The current study combines hyperchaotic excitations and the enhanced capabilities of extended Kalman-Bucy filter to propose a real-time approach for identification of damage in nonlinear structures. Simulation results show that the proposed approach is capable of online identification and assessment of damage in nonlinear elastic and hysteretic structures with single or multiple degrees-of-freedom using noise-corrupted measured acceleration response.

This paper presents a novel time domain approach for Structural Health Monitoring (SHM) systems based on Electromechanical Impedance (EMI) principle and Principal Component Coefficients (PCC), also known as loadings. Differently of typical applications of EMI applied to SHM, which are based on computing the Frequency Response Function (FRF), in this work the procedure is based on the EMI principle but all analysis is conducted directly in time-domain. For this, the PCC are computed from the time response of PZT (Lead Zirconate Titanate) transducers bonded to the monitored structure, which act as actuator and sensor at the same time. The procedure is carried out exciting the PZT transducers using a wide band chirp signal and getting their time responses. The PCC are obtained in both healthy and damaged conditions and used to compute statistics indexes. Tests were carried out on an aircraft aluminum plate and the results have demonstrated the effectiveness of the proposed method making it an excellent approach for SHM applications. Finally, the results using EMI signals in both frequency and time responses are obtained and compared.

The wavelet transform has proven to be a useful mathematical tool to detect changes in the mode shapes of a structure and therefore to detect damage. The authors have proposed a damage detection methodology based on the wavelet analysis of the difference of mode shapes corresponding to a reference state and a potentially damaged state. The wavelet coefficients of each mode shape difference are added up to obtain an overall graphical result along the structure. The coefficients are weighted according to changes in natural frequencies to emphasize the mode shapes most affected by damage. This paper is focused on the enhancement of the damage sensitivity of the methodology. It presents new results when applying a curve fitting approach to reduce experimental noise effect in mode shapes as well as a interpolation technique to virtually increase the geometric sample frequency of the wavelet transform input signal. The enhanced methodology is applied to experimentally tested steel beams with different crack location and depth. The paper analyses the results when considering different number of measuring points. Successful results are obtained using a small number of sensors and mode shapes.

The merit of linear projections as a way to improve the resolution in damage detection under changing environmental conditions is examined. It is contended that if the data from the reference condition is balanced, in the sense that the number of feature vectors available for the various temperatures is similar, then projections, such as those in Principal Component Analysis and Factor Analysis, will not improve performance. Projections, however, help to control the false positive rate when the reference data set is not balanced. Analysis and simulation results suggest that previous claims on the merit of projection as a way to improve damage detection resolution under environmental variability may be too optimistic.

An eigenmode based model reduction technique is proposed to obtain low-order models which contain the dominant eigenvalue subspace of the full system. A frequency-limited interval dominancy is introduced to this technique to measure the output deviation caused by deflation of eigenvalues from the original system in the frequency range of interest. Thus, the dominant eigensolutions with effective contribution can be identified and retained in the reduced-order model. This metric is an explicit formula in terms of the corresponding eigensolution. Hence, the reduction can be made at a low computational cost. In addition, the retained low-order model does not contain any uncontrollable and unobservable eigensolutions. The performance of the created reduced-order models, in regard to the approximation error, is examined by applying three different input signals; unit-impulse, unit-step and linear chirp.

At present, the common algorithm of computing FRF is averaging method in frequency domain. This algorithm is precise method for impact exciting or burst random exciting, but not for continuous exciting, for the initial response cannot be considered. In continuous exciting test, increasing data length which permits increasing averaging times is needed to alleviate the error of FRF caused by the initial response. The initial response is caused by the previous frame of exciting force. Thus, in this work an algorithm model is put forward by considering the initial response. For each averaging computation in frequency domain, the data of two frames force and one frame response, aligned in right end, are used. The initial response is caused by the first frame force. When the FRFs of MISO are known, the IRFs (Impulse Response Function) are obtained by IFFT transform of FRFs. The theoretical response of this point except the first frame can be computed out by the convolution of forces and IRFs. The RMS of error series between theoretical response and measured response, divided by the RMS of measured response, reflects the preciseness of FRF. The smaller is the value, the better. The speed of common averaging method in frequency domain is fast, but with bad FRF preciseness when data is short. The FRF preciseness of least square devolution method in time domain is best, but with the slowest computation speed and unpractical. The preciseness of FRF with new algorithm is very near to the devolution method but the computation time can be shortened greatly. Applying the new algorithm in continuous exciting MIMO test, the test time can be greatly shortened. The new algorithm can also be applied to impact MIMO test, with multi impacts acting in different points at the same time. In the paper, real test and simulating data are used to verify the new algorithm, and the new algorithm is also compared with the time domain iteration method which is put forwarded before.

Model reduction is a technique commonly used to reduce the computation time of structural dynamic models. The results of the reduced model can then be expanded to full space using the transformation matrix developed in the reduction process. This work focuses on expanding the mode shapes of system models that are comprised of reduced component models. Typically, this expansion requires the full space system model to be computed in order to obtain the system model transformation matrix needed to perform the expansion. Computing the full space system model to determine the expansion matrix does not save computation time and therefore defeats the purpose of using model reduction. This paper proposes using the expansion matrices of the individual components to expand the assembled system model modes. In this work, System Equivalent Reduction Expansion Process (SEREP) is used for reduction and expansion. The accuracy of the expanded system model is shown to be dependent on the modes retained in the reduced component models. Recent work on Variability Improvement of Key Inaccurate Node Groups (VIKING) has shown that over specifying the number of modes used in the reduction/expansion process that span the space of the system model modes significantly improves the results. The VIKING technique is the basis for the expansion process developed in this work. Multiple analytical cases are presented to show how the selection of component modes affects the expansion results. The analytical cases demonstrate that accurate system model expansion results can be obtained when a sufficient set of component modes that span the space of the system model modes are used.

A recently introduced theorem on the localization of damage from changes in flexibility is reviewed. The theorem states that the image of the change in the flexibility matrix resulting from damaged is a basis for the influence lines of the stress resultants at the damaged locations. The theorem is the dual of the damage locating vector theorem, which states that the vectors in the kernel lead to stress fields that are zero over closed regions that contain the damage. Extension of the theorem to cases where flexibility matrices cannot be extracted from vibration data is realized by replacing the flexibility change with a surrogate. Performance of the image based localization approach is examined in simulations and in an experimental setting.

This paper presents a predictive model of a cable-harnessed structure through using the Spectral Element Method (SEM) and this is compared to a finite element approach. SEM is an element-based method that combines the generality of the finite element method with the accuracy of spectral techniques. The exact dynamic stiffness matrices are used as the element matrices in the Finite Element Method (FEM). Thus it is possible to generate the meshes on geometric domains of concern. The spectral element can be assembled in the same terminology of the FEM. After assembly and application of the boundary conditions, the global matrix can be solved for response of model, repeatedly at all discrete frequencies because the dynamic stiffness matrices are computed at each frequency. Here we model a cable-harnessed structure as a double beam system. The presented SEM model can define location and number of connections very conveniently. Comparison is conducted between the FEM and SEM for several cases. The results show that the proposed SEM approach can be used as the exact solution of cable harnessed structures.

This paper analyzes the dynamic characteristics of a rolling tire with a modal approach. A ring model is adopted to model a tire and this model is substituted in a general transfer function and considers the rolling condition to make a transfer function of a rolling tire in the Laplace domain and an corresponding impulse response function in the time domain. From this transfer function it is clearly confirmed that the shifting effect, the so-called Doppler Effect, shifts not only the natural frequencies but also the damping ratios.

A procedure for the nonlinear modal analysis of a resistor is presented in this paper. The identification process aims at assessing the viscous damping factor of the structure including the nonlinear terms in the frequency response function with the harmonic balance method. For this purpose a simplified single DOF system has been adopted to describe the nonlinear behavior. A good agreement between numerical and experimental results has been achieved.

In milling and turning processes high speed rotation is used to decrease operation time; as a result, the cost of machining operation is as well reduced. In order to estimate the stability lobes and determine optimum cutting conditions, tool point Frequency Response Function (FRF) is required. Euler and Timoshenko beam models are employed in literature in order to obtain tool point FRF of spindle-holder-tool assemblies. However, due to the high speed rotation of spindle, it is also necessary to consider gyroscopic effects for the determination of tool point FRF. In this paper, spindle-holder-tool assembly is modeled by Timoshenko beam theory considering gyroscopic effects as well. Considering the analytical modal solution available in literature and structural coupling methods, FRF of the tool point is obtained. A case study is performed on an example of spindle-holder-tool assembly, which is as well modeled by finite elements in order to verify the results obtained by the continuous model. Next, using the continuous model, stability lobes are obtained for different spin speeds and the effect of gyroscopic forces are studied.

In this paper, the effect of piezoelectric shunt damping (PSD) on chatter vibrations in turning process is studied. Chatter is a self-excited type of vibration that develops during machining due to process-structure dynamic interactions resulting in modulated chip thickness. Chatter is an important problem, since it results in poor surface quality, reduced productivity and reduced tool life. The regenerative chatter results from phase differences between two subsequent passes of the cutting tool and occurs earlier than mode coupling chatter in most cases. In regenerative chatter theory, stability limit in the cutting process is inversely proportional to the negative real part of frequency response function (FRF) of the cutting tool-workpiece assembly. If the negative real part of the FRF at the cutting point can be decreased, depth of cut, in other words productivity rates, will increase. In piezoelectric shunt damping method, an electrical impedance is connected to a piezoelectric transducer which is bonded to the main structure. In this study, resistive – inductive – capacitive shunt circuit is used, whose elements are optimized with genetic algorithm to minimize the real part of the FRF for certain target frequencies. Afterwards, the effect of the optimized piezoelectric shunt damping on the absolute stability limit of the cutting process is investigated.

On the premise that damage is localized, as opposed to distributed, it follows that difference in the displacement field between the damaged and the reference state due any loading in sensor coordinates is in a subspace whose span is defined by the influence lines of the stress resultants at the damage locations. It is shown that the scaling factors for which the bases vectors combine to give the displacement difference are the amplitude of discontinuities collocated with the damage and that these discontinuities, together with the stress resultant at the same locations, provide means to evaluate the damage severity. The solution proves to be an attractive fixed point.

As wind turbine blades fatigue, the blade’s dynamic response to loading may be expected to change. The kinematic quantities that exhibit significant changes are important for wind turbine blade operation from the perspective of measurement, estimation, and performance or even life cycle prediction. A state estimate providing accurate information on these features would lead to better estimates of remaining fatigue life and provide valuable information to the turbine control systems for the purpose of maximizing total energy output of a wind turbine system. In this work, we implement an observer for state estimation of nonlinear systems using the system Jacobian to correct the system output by updating the force input to a reference model in an iterative Newton-Raphson scheme. We apply this method to a surrogate wind turbine blade modeled using a geometrically exact beam theory to estimate its state given available measurements.

LA-UR-12-25441

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As novel construction materials become more economically and technically available throughout the globe, the structures become more complex. Therefore, an unexpected failure of these structures causes catastrophic economical and fatal losses. Over the past decades, a number of vibration based damage detection techniques have been developed to avoid unexpected failure in structures. These damage detection methods identify presence, location and magnitude of developed damage in the structure associated to change of modal properties such as natural frequency, mode shape and modal damping. An essential limitation of these techniques is that they are majorly sensitive to high intensity of damage only and do not correctly respond to minor to intermediate damage severities. This research proposed a novel damage detection methodology based on application of Hilbert-Huang Transform (HHT) on the acceleration response of the structure. HHT consists of Empirical Mode Decomposition followed by Hilbert Transform of the signal. The method develops a Damage Index Matrix for the structure by connecting energy index of the instrumented structural points. Viability of the proposed method is demonstrated through numerical examples and laboratory experiments. The method was able to locate both single and multiple damage scenarios in numerical models as well as laboratory experiment.

The present report is devoted to present an approach to dynamic analysis of multiple cracked Euler-Bernoulli beam subjected to general moving load. The novelty of the approach consists of using analytical solution of vibration mode of multiple cracked beam in the frequency domain that is straightforward to compute the time response of multiple cracked beam to moving load given generally in a discrete form. The proposed method enables to eliminate the “moving singularity” phenomena that trouble the use of either the conventional modal method or the modern numerical techniques. The theoretical development is illustrated by numerical results.

Structural damages usually introduce nonlinearity to the system. A previously developed nonlinear identification method is employed to detect crack type structural damage. The method requires the measurement of FRFs at various points in order to locate the damage. The method makes it also possible to determine the extent of damage by identifying the level of nonlinearity. The verification of the method is demonstrated with experimental case studies using beams with different levels of cracks. The approach proposed in this study is very promising to be used in practical systems, but still open to further improvements.

In this study, vibration fatigue analysis of a cantilever beam is performed using an in-house numerical code. Finite element model (FEM) of the cantilever beam verified by tests is used for the analysis. Several vibration fatigue theories are used to obtain fatigue life of the cantilever beam for white noise random input and the results obtained are compared with each other. Fatigue life calculations are repeated for different damping ratios and the effect of damping ratio is studied. Moreover, using strain data obtained from cantilever beam experiments, fatigue life of the beam is determined by utilizing time domain (Rainflow counting method) and frequency domain methods, which are compared with each other. In addition to this, fatigue tests are performed on cantilever beam specimens and fatigue life results obtained experimentally are compared with that of in-house numerical code. It is observed that the accuracy of the damping ratio is very important for accurate determination of fatigue life. Furthermore, for the case considered, it is observed that the fatigue life result obtained from Dirlik method is considerably similar to that of Rainflow counting method.

This paper presents a newly developed method for obtaining the modal model with a proper model order from experimental frequency response functions (FRF). The method is a multi-step procedure which commences with the identification of a high-order state-space model, Exhaustive Model (EM), using the full FRF data set. Then, modal states that give small contribution to the output, quantified by a metric associated to the observability grammian, are rejected from the EM resulting in a Reference Model (RM). Competing models, with the same model order as the RM, are then found by bootstrapping realization using same-size fractions of the full FRF. Eigensolutions of the Bootstrapping Models (BMs) are then paired by the eigensolutions of the RM based on high Modal Observability Correlation (MOC) indices. In a second reduction stage, the modal states with low MOC index are rejected from the BMs. Final model is found by an averaging through BMs. Only one threshold quantity, related to observability grammians need to be set by the user. The method thus requires very little user interaction. The method is applied to experimental data used in a previous IMAC Round Robin exercise for experimental modal analysis evaluation.

The historical development of the Modal Assurance Criterion (MAC) originated from the need of a correlation metric for comparingexperimental modal vectors, estimated from measured data, to eigenvectors that have been determined from finite element calculation. For systems with well separated eigenvalues with many system degrees-of-freedom (DOF) represented in the eigenvectors it is normally easy to distinguish eigenvectors associated to different eigenvalues by low MAC correlation numbers. However, for eigenvectors with a sparse DOF sampling it may be hard to distinguish between vectors by MAC correlation numbers. To reduce the problem of distinguishing between eigensolutions, this paper advocates the use of a new correlation metric based on the observability matrix of the diagonal state-space realization. This is instead of using a metric based on the eigenvectors only.

A modal test method that uses sound pressure transducers at fixed locations and an impact hammer roving over a test structure is developed in this work. Since sound pressure transducers are used, the current method deals with a coupled structural-acoustic system. Based on the vibro-acoustic reciprocity, the method is equivalent to the one, where acoustic excitations at fixed locations are given and the resulting acceleration of the test structure is measured. The current method can eliminate mass loading due to the use of accelerometers, which can destroy the existence of repeated or close natural frequencies of a symmetric structure, avoid the effects of a nodal line of a mode and an inactive area of a local mode, and measure all the out-of-plane modes within a frequency range of interest, including global and local ones. The coupling between the structure and the acoustic field in a structural-acoustic system introduces asymmetry in the model formulation. An equivalent state space formulation is used for a structurally damped structural-acoustic system and the associated eigenvalue problem is derived. The biorthonormality relations between the left and right eigenvectors and the relations between the structural and acoustic components in the left and right eigenvectors are proved. The frequency response functions associated with the current method are derived and their physical meanings are explained. The guidelines for using the current method, including the types of structures that are suitable for the method, the positions of the sound pressure transducers, and the orientation of the test structure relative to the transducers, are provided. Modal tests were carried out on an automotive disk brake using the traditional and current methods, where multiple accelerometers and microphones were used to measure its dynamic responses induced by impacts, respectively. The measured natural frequencies and mode shapes by the two methods are almost the same. The differences between the measured natural frequencies using the current method and those from the finite element model of the disk brake are less than 3% for the first 18 elastic modes, and the modal assurance criterion values of the associated mode shapes are all above 90%.

Due to more stringent legal sound regulations and customer demand of quieter products, the acoustic requirements for product design become a major task in the product development. For the requirement for lighter structures, synthetic materials like glass fibers become more prominent in recent designs. For high accurate simulation corresponding material data are needed. But concerning isotropic plastics the Young’s modulus and damping are highly frequency- and temperature-dependent. A novel test to determine the frequency- and temperature-dependent Young’s modulus and damping of isotropic plastics will be shown in this paper. With a special test facility a sample is excited reactionless by a constant repeatable force during measuring the surface velocity. A novel contactless sample suspension is developed. The modal parameters, response frequency and damping, are extracted from these measurements. With the assistance of FEA simulations the Young’s modulus can be adapted step by step to these modal parameters. The Young’s modulus can be extracted frequency- and temperature-dependent.

In this study, we introduce a real-time vision-based modal testing system to estimate the modal parameters of shell-shaped objects simultaneously using high-frame-rate structured-light-based vision. Based on structured-light-based vision consisting of a DLP LightCrafter projector and a high speed vision platform (IDP Express), the transient three-dimensional (3D) geometry of a vibrating shell-shaped object can be measured in real time at 1,000 fps as the 3D vibrational displacements of 100 points on an object in the audio frequency range. In our system, the modal parameters are estimated simultaneously for the 3D displacements of the shell-shaped object using a fast output-only modal parameter estimation algorithm, SSI-CPAST, on the IDP Express. Therefore, our modal testing system can simultaneously monitor and inspect the input-invariant structural dynamic properties of vibrating shell-shaped objects excited at dozens or hundreds of Hertz. Our system can be applied to real-time vision-based structural damage detection and fatigue testing in various applications. To demonstrate the performance of our vision-based modal testing system, an experiment was performed in real time to estimate the resonant frequencies and mode shapes of steel plates with artificial damage, which were excited at dozens or hundreds of Hertz using a hammer.

Surface mining machines are the largest mechanical engineering structures. Bucket wheel excavators operating in lignite mines are continually exposed to dynamic loads. Moreover, nearly all structures are over 10 years old. The methods used in design and construction did not cover the dynamic behavior of machines, which resulted in problems in their operation and decreased durability. The tendency to optimize and increase the operational time of machines is currently also visible in the field of surface mining. As a result the machines which have been operating for many years are subject to investigation. The SchRs (Schaufelradbagger auf Raupenschwenkbar) 4600.50 excavator, which is over 120 m long, 64 m high and weighs approximately 5,000 tons (excluding the dumping bridge), was investigated with regard to vibrations. Operational modal analysis was performed with 15 measurement points in different directions. This approach allowed to determine the variability of the dynamic characteristics of the machine in terms of operational conditions. Simultaneously a numerical model was prepared. Eventually, it was possible to compare and correlate both the numerical and experimental models and to establish the differences based on the operational load acting on the structure.

This study presents an operational modal analysis of rotating Carbon Nanotube (CNT) infused composite beams in order to explore the effect of CNT’s on the natural frequencies and damping characteristics of the composite structure during rotation. Engineering applications with rotating components made from composites often suffer from excess vibrations because of the inherent high stiffness to weight ratio of the composite material and the oscillating loads from rotation. Previous research has demonstrated that the addition of CNT’s to composite resins increases the damping characteristics of the resulting material, and several of these works have suggested that CNT-infused composites may be useful in rotor design as a means of passive vibration suppression. The present work aims to address this suggestion with an experimental investigation using composite beams fabricated with CNT’s embedded in an epoxy resin matrix along with several layers of reinforcing carbon fiber fabric. An experimental apparatus is designed and constructed to hold two cantilever composite beams on opposite sides of a rotating central shaft controlled via a DC servo motor and a PID control loop. White noise is generated and added to the input motor RPM signal to randomly excite the base of the structure during rotation, and the Eigensystem Realization Algorithm (ERA) is used to analyze the data measured from the vibrating beam in order to determine the modal parameters of the system. The extracted modal parameters are presented as a function of the angular speed and weight percentage CNT loading in order to gain insight into application areas involving vibration suppression in rotating composite structures such as helicopter rotors and wind turbine blades.

Heliostats are structures that track the sun and reflect sunlight to a centrally located receiver on top of a tower to produce heat for electricity generation. Commercial power towers can consist of thousands of heliostats that are subject to wind-induced loads, vibration, and gravity-induced sag. This paper presents modal tests of a heliostat located at the National Solar Thermal Test Facility (NSTTF) at Sandia National Labs in Albuquerque, New Mexico. The heliostat was instrumented with 22 accelerometers, 4 strain gauges, and 3 wind anemometers to examine manually and wind-induced vibrations of the structure. Data acquisition software was developed to provide real-time monitoring of the wind velocity, heliostat strain, mode shapes, and natural frequencies which will be used to validate finite element models of the heliostat. The ability to test and monitor full-scale heliostats under dynamic wind loads will provide a new level of characterization and understanding compared to previous tests that utilized scaled models in wind-tunnel tests. Also, the development of validated structural dynamics models will enable improved designs to mitigate the impacts of dynamic wind loads on structural fatigue and optical performance.

High-speed milling operations of thin walls are often limited by the so-called regenerative effect that causes poor surface finishing. To optimize the cutting process in terms of quality surface and productivity, the frequency response function (FRF) of the wall needs to be measured in order to identify the modal parameters of the system which are used to obtain the stability lobes that help identify the optimal system’s parameter values to warrant stable cutting conditions. The aim of this work is to experimentally show the variation on the frequency response function (FRF) values obtained by using a laser Doppler vibrometer (LDV) device and accelerometer sensors during a milling operation processes of an aluminum thin-walled workpiece of 1 mm thick and 30 mm height. It is shown that the FRF values variations has strong influence on the stable cutting bounds. To further assess our findings, we used the collected experimental data obtained by using the LDV during milling machine cutting operation processes of several thin-walled workpieces to identify the cutting parameters values that allow us to obtain good quality and acceptable surface finish.

The present study focuses on Model Order Reduction (MOR) methods of non-intrusive nature that can be seen as belonging to the category of system identification techniques. Indeed, whereas the system to analyze is considered as a black box, the accurate modeling of the relationship between its input and output is the aim of the proposed techniques. In this framework, the paper deals with two different methodologies for the system identification of thermal problems. The first identifies a linear thermal system by means of an Extended Kalman Filter (EKF). The approach starts from an a priori guessed analytical model whose expression is assumed to describe appropriately the response of the system to identify. Then, the EKF is used for estimating the model transient states and parameters. However, this methodology is not extended to the processing of nonlinear systems due to the difficulty related to the analytical model construction step. Therefore, a second approach is presented, based on an Unscented Kalman Filter (UKF). Finally, a Finite Element (FE) model is used as a reference, and the good agreement between the FE results and the responses produced by the EKF and UKF methods in the linear case fully illustrate their interest.

Online monitoring of modal and physical parameters which change due to damage progression and aging of mechanical and structural systems is important for the condition and health monitoring of these systems. Usually, only the limited number of imperfect, noisy system state measurements is available, thus identification of time-varying systems with nonlinearities can be a very challenging task. In order to avoid conventional least squares and gradient identification methods which require uni-modal and double differentiable objective functions, this work proposes a modified differential evolution (DE) algorithm for the identification of time-varying systems. DE is an evolutionary optimisation method developed to perform direct search in a continuous space without requiring any derivative estimation. DE is modified so that the objective function changes with time to account for the continuing inclusion of new data within an error metric. This paper presents results of identification of a time-varying SDOF system with Coulomb friction using simulated noise-free and noisy data for the case of time-varying friction coefficient, stiffness and damping. The obtained results are promising and the focus of the further work will be on the convergence study with respect to parameters of DE and on applying the method to experimental data.

Mechanical connections play a significant role in predicting the dynamic characteristics of assembled structures accurately. Therefore, several methods were developed to determine equivalent dynamic models for joints. In this paper an experimental identification method based on FRF decoupling and optimization algorithm is proposed for modeling structural joints. The method developed is an extension of the method proposed by the authors in an earlier work. In the method proposed in our earlier work FRFs of two substructures connected with a bolted joint are measured, while the FRFs of the substructures are obtained analytically or experimentally. Then the joint properties are calculated in terms of translational, rotational and cross-coupling stiffness and damping values by using FRF decoupling. In this present work, an optimization algorithm is used to update the values obtained from FRF decoupling. The validity and application of the proposed method are demonstrated with experimental case studies. Furthermore, the effects of bolt size on joint dynamics are also studied by making a series of experiments and identifying the joint parameters for each case.

A laser Doppler vibrometer typically measures the translational velocity at a single point along the direction of incident light. However, it has been shown that rotational velocities can also be recovered by scanning the laser continuously along a line or circular path around that point. This work uses the harmonic transfer function concept, which is analogous to the transfer function in conventional modal analysis, to relate the measured rotational and translational velocities to the input force. With this concept, the continuous-scan approach can be combined with the conventional point by point scan approach, acquiring normalized translational and rotational velocities under various types of excitation conditions in the same amount of time that is required for obtaining only the translational velocity. The proposed approach is validated on measurements taken from a downhill ski under free-free boundary conditions. The influence of the circle size, the scanning rate and the surface quality on the noise level in the measured signal is discussed, and the measured deflection shapes using both the point and circular scan approaches are compared. Local slopes at measurement locations are computed from the identified principal rotational velocities, laying the foundation for constructing a much more accurate estimate of the deformation shape, which may be valuable in damage detection and/or model updating.

We utilize the nonlinear system identification (NSI) methodology, which was recently developed based on the correspondence between analytical and empirical slow-flow dynamics. Performing empirical mode decomposition on the simulated or measured time series to extract intrinsic mode oscillations, we establish nonlinear interaction models, which invoke slowly-varying forcing amplitudes that can be computed from empirical slow-flows. By comparing the spatio-temporal variations of the nonlinear modal interactions for structures with defects and those for the underlying healthy structure, we will demonstrate that the proposed NSI method can not only explore the smooth/nonsmooth nonlinear dynamics caused by structural damage, but also the extracted vibration characteristics can directly be implemented for structural health monitoring and detecting damage locations. Starting with traditional tools such as the modal assurance criterion (MAC) and the coordinate MAC are utilized.

The work herein presents a novel approach for sound field characterization based on a continuous scanning/roving approach. Conventional methods use sensor arrays or a discreet moving sensor to characterize an acoustic field. Although these methods are well established, this new approach attempts to take advantage of a continuous measurement methodology, in space and time, in order to increase spatial resolution of the acoustic field, minimize the use of sensors required, and the acquisition time. The novel approach relies on processing amplitude-modulated time signal, with geometrical reference, in order to characterize the acoustic field. This technique can be thought of as an extension of continuous scanning laser Doppler vibrometry techniques.The work is demonstrated on a lab test article and used to identify the acoustic propagation generated by the excitation of several structural modes. The work shown here is preliminary and mainly aimed at proving the methods feasibility.

The action of bowing a violin string to play a properly sounding note produces a stick-slip action between the bow and the string. In an ideal situation this excitation of the string results in Helmholtz motion of the string. This paper describes an experimental method for measuring the motion of a violin string using high speed/high resolution videography and the subsequent digitizing of the motion for use in a high fidelity mathematical model of a violin.

Full field measurement techniques can provide fast, accurate and detailed vibration response data for finite element model validation and are being wildly used in industrial applications. A Scanning Laser Doppler Vibrometer (SLDV) measures the full field operating deflection shapes of a structure by changing the location of the laser spot on the target surface. However the setup of a measurement and in particular the measurement grid definition, can take up a significant part of the overall measurement time. To optimise the setup time a novel technique for SLDV measurement grids has been developed. The suggested method includes an automated identification of the vibrating target, based on the measured vibration signal of a scan covering the entire field of view of the LDV. Alpha-shape techniques for target identification and geometric algorithms for shape recognition are used to define the measurement area. Novel approaches for symmetry and orientation capture allow the generation of point grids and continuous patterns for various target shapes. The introduced approach allows a quick SLDV setup of the full field scan with a minimum input from the user.

The paper illustrates the idea of using theoretical knowledge of specific structure’s mode shapes as a way of filtering time domain data obtained by Continuous Scanning Laser Doppler Vibrometry (CSLDV). The CSLDV output measures the structure vibration joining together the spatial and time information. It is proposed here to exploit the a priori knowledge of the candidate mode shape spatial distributions to extract from the vibration data the resonance frequency information with high accuracy. That technique is based on the concept that modal analysis ends up with a final abstraction and labelling of the mode shapes on the basis of their nodal lines position, i.e. first, second bending and/or torsional, etc. The expected mode is compared with the experimental data in order to evidence the information on the frequency at which that mode occurs.

In the last years, simulations have been extensively used to study mechanical systems. However, sometimes mathematical models do not adequately describe the behavior of some system components and, in some cases, only these components need to be tested. Thus, the Hardware In the Loop (HIL) technique was developed to solve these problems, allowing a real time simulation of a hybrid system composed of physical and virtual parts. Due to this characteristic and the potential cost reduction capacity, without loss of precision and reliability, the HIL technique becomes an important tool for researchers and engineers in different areas such as the automotive industry, robotics and production in general. However, the result shows that there is a time delay phenomenon between the expected and obtained signal. This paper analyzes the time delay effects in a HIL simulation by means of an example of a mass-spring-damper system. The analysis results shows that the time delay comes from the actuator dynamic response and according to the sample time used in the real time system, it strongly affects the HIL simulation transient accuracy and stability. To obtain the same result obtained in the pure simulation, a time delay compensation is applied in the model and its effectiveness was verified by a comparison between full simulated and experimental HIL results.

Piezoelectric materials have proven themselves to be used as actuators for active vibration control. In this study, the active vibration control of a cylindrical shell by means of piezoelectric actuators is investigated. As the most important stage of controlling the vibrations, this paper focuses on the optimal placement of piezo patch actuators. It is aimed in this study to determine the optimum locations of piezo patches on a cylindrical shell by using finite element model of the system. In optimization, the spillover effects, which are caused by accidentally excited higher modes of the structure are also taken into account.

Active control systems are used on aircraft to reduce loads due to gusts and manoeuvres, reduce the effect of noise, and could also increase the speed at which flutter occurs. Unfortunately most aeroservoelastic systems include some form of nonlinearity, and this increases the complexity of the feedback system and also facilitates the likelihood of Limit Cycle Oscillations occurring. Previous work on the application of Adaptive Feedback Linearisation to aeroelastic systems has demonstrated the promising potential of this method when applying control in the presence of substantial nonlinearity. In this work, Adaptive Feedback Linearisation is applied to an aeroelastic model of a cantilevered flexible wing with a cubic hardening structural nonlinearity in an engine pylon. Using assumed vibration modes, a suitable model of the wing is developed, into which structural nonlinearity is incorporated. Closed-loop control is implemented on the aeroservoelastic system via linearising feedback computed through the Adaptive Feedback Linearisation algorithm. The advantage of the latter is the guaranteed stability of the closed-loop aeroelastic system, despite lack of knowledge of the exact description of the nonlinearity. It is shown how such an approach can be used to delay the onset of flutter or limit cycle oscillations.

The problem of flutter suppression in aeroelasticity may be treated using eigenvalue assignment. Thus the receptance method of eigenvalue assignment developed by Ram and Mottershead may be considered for flutter suppression. Also, nonlinear flutter is characterized by periodicity. Therefore, the describing function technique is applicable to quasi-linearise the nonlinear aeroelastic system such that traditional linear methods of analysis can be employed. Eigenvalue assignment is especially relevant in the case of LCOs, which are neutrally stable and therefore are readily assignable in the frequency domain. In this paper, structural nonlinearities will be considered. Approximating structural nonlinearities using describing functions, the theory of the nonlinear active control based on the receptance method will be achieved by extending the well-developed theory of linear active control. Also, a new limit cycle prediction method in the frequency domain and a new form of stability criterion of limit cycles for closed-loop systems based on the describing function method will be derived. Numerical results from two two-degree-of-freedom airfoils with cubic and piecewise nonlinear pitching stiffnesses respectively are presented to show that limit cycles can be predicted precisely by the new method and the proposed nonlinear active control technique might be able to suppress flutter into stable LCO.

Interior noise inside the passenger cabin of ground vehicles can be classified as structure-borne and airborne. The disturbance caused by the engine forces excites the panels enclosing the passenger cabin to vibrate at their resonance frequencies. These vibrating panels cause changes in the sound pressure levels within the passenger cabin, and consequently generating an undesirable booming noise. In this study, we developed a methodology to design an active structural acoustic controller (ASAC) to attenuate the structure-borne noise, which is mainly caused by the most influential radiating panel. The panel is determined by conducting panel acoustic contribution analysis (PACA) based on the acoustic transfer vector (ATV) methodology. Then, active structural acoustic controller is designed for vibration suppression of this panel, which has complex geometry and boundary conditions. The performance of the controller for noise reduction is investigated for various controller parameters and sensor/actuator positions. It is shown that an optimization algorithm is required to determine the optimum controller parameters and sensor/actuator positions to reduce sound pressure levels inside the cabin efficiently.

In the context of this paper, a small scale, medium precision, stabilized pan/tilt platform is developed as a prototype, which is used to compare various stabilization algorithms experimentally. The overall performance of the system depends on rigid body dynamics, structural dynamics, servo control loops, stabilization control algorithm, sensor fusion algorithm, and sensory feedback such as from the inertial measurement unit (IMU). In the case that the response bandwidth of the overall system is high enough, the same hardware can also achieve active vibration isolation. All of these design aspects are investigated in the paper via numerical models and with their experimental verification.

Dynamic equations for an anisotropic cylindrical shell are derived using a series expansion technique. First the displacement components are expanded into power series in the thickness coordinate relative to the mid-surface of the shell. By using these expansions, the three-dimensional elastodynamic equations yield a set of recursion relations among the expansion functions that can be used to eliminate all but the six of lowest order and to express higher order expansion functions in terms of these of lowest orders. Applying the boundary conditions on the surfaces of the cylindrical shell and eliminating all but the six lowest order expansion functions give the shell equations as a power series in the shell thickness. These six differential equations can in principle be truncated to any order. The method is believed to be asymptotically correct to any order. For the special case of a ring, the eigenfrequencies are compared with exact two-dimensional theory, generally with a good correspondence.

Finite element models of structural systems have become increasingly complicated as they are used for many types of applications. Because a highly detailed model is not required for every purpose, much work has been presented in generating reduced-order models (ROMs) that accurately reflect the dynamic characteristics of the system. Recent work has shown that these ROMs can be used to compute both the linear and nonlinear dynamic response of the structure at the reduced space without loss of accuracy due to the reduction procedures.

Although computationally beneficial to calculate dynamic response at drastically reduced space, there are significant advantages in expanding the response out to the full-field solution. Much work has been focused on the expansion of linear dynamic solutions for improved visualization. This work builds on previous efforts and demonstrates that the transformation matrix generated from the linear model can be used to accurately expand the nonlinear dynamic response; this is true for component as well as system models. Both analytical and experimental cases are examined to demonstrate the accuracy of this technique.

In this paper we present a new method for constructing one-dimensional vibrating systems having prescribed values of the first N natural frequencies, under a given set of boundary conditions. In the case of axially vibrating rods, the analysis is based on the determination of the so-called quasi-isospectral rods, that is rods which have the same spectrum as a given rod, with the exception of a single eigenvalue which is free to move in a prescribed interval. The reconstruction procedure needs the specification of an initial rod whose eigenvalues must be close to the assigned eigenvalues. The rods and their normal modes can be constructed explicitly by means of closed-form expressions. The results can be extended to strings and to some special classes of beams in bending vibration.

The strength of the modal based reduction approach resides in its simplicity, applicability to treat moderate-size systems and also in the fact that it preserves the original system pole locations. However, the main restriction has been in the lack of reliable techniques for identifying the modes that dominate the input-output relationship. To address this problem, an enhanced modal dominancy approach for reduction of second-order systems is presented. Briefly stated, a modal reduction approach is combined with optimality considerations such that the difference between the frequency response function of the full and reduced modal model is minimized in

$$\mathcal{H}_{2}$$

sense. A modal ranking process is performed without solving Lyapunov equations. In the first part of this study, a literature survey on different model reduction approaches and a theoretical investigation of the modified modal approach is presented. The error analysis of the proposed dominancy metric is carried out. Furthermore, the performance of the method is validated for a lightly damped structure and the results are compared with other dominancy metrics. Finally the optimality of the obtained reduced model is discussed and the results are compared with the optimum solution.

In the first part of this study, a theoretical investigation of an improved modal approach and a complete error analysis of the proposed modal dominancy metric were presented. In this part the problem of metric non-uniqueness for systems with multiple eigenvalues is described and a method to circumvent this problem based on spatial distribution of either the sensors or the actuators is proposed. This technique is implemented using QR factorization without solving Lyapunov equations. Moreover, the method is improved such that it is able to use the information extracted from spectral properties of the input. Also in order to make the method more effective, information extracted from the input internal structure is incorporated in the modal ranking process. It is shown that this improvement is particularly effective in problems with high-dimensional input and/or output space such as in distributed loading and moving load problems. Finally the performance of the method is validated for a high order system subjected to a high-dimensional input force. That originates from a railway track moving load problem.

Abstract In this paper, a review of Gramian-based minimal realization algorithms is presented, several comments regarding their properties are given and the ill-condition and efficiency that arise in balancing of large-scale realizations is being addressed. A new algorithm to treat non-minimal realization of very large second-order systems with dense clusters of close eigenvalues is proposed. The method benefits the effectiveness of balancing techniques in treating of non-minimal realizations in combination with the computational efficiency of modal techniques to treat large-scale problems.