Mode Shape Components Research Articles

Online operational modal decomposition

This paper presents a novel approach for online operational modal decomposition utilizing nonstationary noisy measurements only. Critical problems include the online updating of modal participation factors and the estimation of mode shape components and they are successfully resolved in the proposed online operational modal decomposition framework. The algorithm is performed in a two-phase manner. In the first phase, mode shape components for the observed degrees of freedom are extracted from the nonstationary noisy measurements. In the second phase, modal coordinates are updated recursively by combining a modal extended Kalman filter and a Bayesian probabilistic approach for online updating the modal participation factors. The proposed algorithm requires neither a finite element model of the dynamical system nor the assumptions about stationarity of the measurements and, thus, it provides a computationally efficient framework for online operational modal decomposition of dynamical systems. An illustrative example and a practical application are utilized to demonstrate the applicability and efficiency of the proposed approach.

Reconstruction of radial pseudo-forces on cylinders via mono-laser scanning: Concept and application to characterization of damage

Damage in a plate can cause perturbation to the equation of out-of-plane motion, which can be equivalently regarded as the transverse pseudo-force applied on the damage region. Thereby, the characterization of damage in the plate can be achieved by identifying the pseudo-force. Nevertheless, it is difficult to formulate the pseudo-force for the characterization of damage in a cylinder because its longitudinal, circumferential, and radial motions are coupled. Moreover, a conventional scanning laser vibrometer equipped with a single scanning head can measure the out-of-plane motions of a plate but is insufficient for measuring the 3D motions of a cylinder. The aforementioned limitation leads to a noticeable barrier to extending the pseudo-force approach from a plate to a cylinder. To overcome this barrier, a new concept of radial pseudo-force (RPF) is formulated to represent the damage-induced perturbation to the radial equilibrium of a cylinder, whose motions are dominated by its radial components. The RPF is applied on the lateral surface of the cylinder and is an ideal damage indicator because it appears in the damage region only and almost vanishes at intact locations. Furthermore, a novel method for reconstructing RPFs via mono-laser scanning is proposed. A correction matrix is formed to correct an undeflected-laser-axis mode shape component (MSC) that can be directly measured via mono-laser scanning, whereby the radial MSC of the cylinder can be obtained for the reconstruction of the RPF on the cylinder. The concept of the RPF is numerically proved using the finite element method. In addition, the applicability of the RPF approach is experimentally validated on a cylinder with internal damage. The numerical and experimental results reveal that the RPF approach can graphically characterize the occurrence, location, and approximate plane size of damage in cylinders.

Finite Cross-Section Method for Mode Shape Recognition of Highly Coupled Beam-Type Structures

Abstract The mode shapes of beam-type structures, such as aircraft wings and wind turbine blades, involve a high degree of coupling between bending and torsional deformations. In the case of wind turbine blades, different types of deformation are typically easily recognized by visual observation. However, this visual approach is sometimes challenging for high-order mode shapes, which involve complex coupling of both bending and torsional deformations. This work proposes a novel mode shape recognition algorithm, called the finite cross-section method (FCSM), for application to highly coupled beam-type structures not only to identify the deformation components of complex beam mode shapes but also more importantly to quantify their respective relative contribution. In the application case study for the FCSM, the entire structure is discretized into multiple cross sections. The flap-wise, edge-wise, and torsional deformation components of the entire structure are determined at the cross-section level. The deformation components of the entire structure and their respective contribution are obtained from assembling all cross sections. To validate the mode shape recognition performance, FCSM is applied to and demonstrated on four test cases: (1) numerical mode shapes of a simple cantilever beam, (2) numerical mode shapes from a straight wind turbine blade, (3) numerical mode shapes of a swept wind turbine blade, and (4) experimental mode shapes from a high spatial resolution 3D scanning laser Doppler vibrometer (SLDV) modal test. Both numerical and experimental studies demonstrate that FCSM can successfully recognize the quantitative contribution of flap-wise, edge-wise, and torsional deformation.

T-tail flutter simulations with regard to quadratic mode shape components

It is known that the dynamic aeroelastic stability of T-tails is dependent on the steady aerodynamic forces at aircraft trim condition. Accounting for this dependency in the flutter solution process involves correction methods for doublet lattice method (DLM) unsteady aerodynamics, enhanced DLM algorithms, unsteady vortex lattice methods (UVLM), or the use of CFD. However, the aerodynamic improvements along with a commonly applied modal approach with linear displacements results in spurious stiffness terms, which distort the flutter velocity prediction. Hence, a higher order structural approach with quadratic mode shape components is required for accurate flutter velocity prediction of T-tails. For the study of the effects of quadratic mode shape components on T-tail flutter, a generic tail configuration without sweep and taper is used. Euler based CFD simulations are applied involving a linearized frequency domain (LFD) approach to determine the generalized aerodynamic forces. These forces are obtained based on steady CFD computations at varying horizontal tail plane (HTP) incidence angles. The quadratic mode shape components of the fundamental structural modes for the vertical tail plane (VTP), i.e., out-of-plane bending and torsion, are received from nonlinear as well as linear finite element analyses. Modal coupling resulting solely from the extended modal representation of the structure and its influence on T-tail flutter is studied. The g-method is applied to solve for the flutter velocities and corresponding flutter mode shapes. The impact of the quadratic mode shape components is visualized in terms of flutter velocities in dependency of the HTP incidence angle and the static aerodynamic HTP loading.

Output-only modal analysis for non-synchronous data using stochastic sub-space identification

The stochastic subspace identification (SSI) method has been recognized as the most dominant and popular system identification technique in the time domain. Nevertheless, it cannot cope with the non-synchronicity of dynamic response measurements that happens sometimes in structural health monitoring practice. To overcome this, this study proposes a strategy for the SSI algorithm to realize system identification from non-synchronous dynamic response measurements. The modal identification is carried out in a pairwise manner by pairing a time-shifted signal with a common reference signal. The state space model is employed to fit the pair of reference and time-shifted signals and the SSI algorithm is exploited to extract the modal parameters. The core of the strategy is the tactical usage of the mean phase deviation (MPD) of the mode for seeking out the actual time lag as well as the actual mode shape components simultaneously. By so doing, the strategy fulfills integrated time lag estimation and modal extraction for non-synchronous dynamic response measurements. Furthermore, the strategy also overcomes the difficulty to determine the model order with the use of the stabilization diagram and reduces the computational cost substantially with the help of the periodicity of the MPD. To examine the performance of the strategy, intensive validations are conducted by making use of the non-synchronous acceleration measurements of the Jiangyin Bridge subjected to a ship collision and the artificially misaligned acceleration measurements of the Canton Tower struck by an earthquake. Both the time lags and the mode shapes identified by the strategy can be well validated, indicating that the proposed strategy is competent for modal identification of non-synchronous dynamic response measurements.

Parameter identification for a point-supported cross laminated timber slab based on experimental and numerical modal analysis

In this paper, the dynamic properties of a point-supported cross-laminated timber slab are studied in order to determine the elastic material parameters on this basis. A detailed experimental modal analysis of the slab with dimensions 16.0 m x 11.0 m is performed, and seven modes including the natural frequencies, damping ratios and mode shape components at 651 sensor positions are identified. The found mode shapes are complex due to environmental influences that occurred during the two-day measurement campaign. This error is corrected by eliminating these influences. A finite element model of the slab is presented, whose parameters in terms of material properties and boundary conditions are determined by a model updating procedure. Based on the modal properties of the seven experimentally identified modes, an accurate and robust parameter set is obtained, which can be used in further numerical studies of the considered CLT to check serviceability limit criteria.

Dynamic Response Characterization of Floating Structures Based on Numerical Simulations

Output-only methods are widely used to characterize the dynamic behavior of very diverse structures. However, their application to floating structures may be limited due to their strong nonlinear behavior. Therefore, since there is very little experience on the application of these experimental tools to these very peculiar structures, it is very important to develop studies, either based on numerical simulations or on real experimental data, to better understand their potential and limitations. In an initial phase, the use of numerical simulations permits a better control of all the involved variables. In this work, the Covariance-driven Stochastic Subspace Identification (SSI-COV) algorithm is applied to numerically simulated data of two different solutions to Floating Offshore Wind Turbines (FOWT) and for its capability of tracking the rigid body motion modal properties and susceptibility to different modeling restrictions and environmental conditions tested. The feasibility of applying the methods in an automated fashion in the processing of a large number of datasets is also evaluated. While the structure natural frequencies were consistently obtained from all the simulations, some difficulties were observed in the estimation of the mode shape components in the most changeling scenarios. The estimated modal damping coefficients were in good agreement with the expected results. From all the results, it can be concluded that output-only methods are capable of characterizing the dynamic behavior of a floating structure, even in the context of continuous dynamic monitoring using automated tracking of the modal properties, and should now be tested under uncontrolled environmental loads.

Damage detection in a suspension bridge using modal flexibility method

Advances in structural engineering and material technology have resulted in increasing use of suspension bridges in infrastructure systems to satisfy the modern transportation requirements. However, during the long life of a suspension bridge, its main cables and hangers could suffer severe corrosion and fatigue leading to failure of at least a part of the bridge. There is a need for a simple and reliable procedure to detect and locate such damage at its onset so that appropriate retrofitting can be carried out to prevent the failure of these important structural elements. Current Structural Health Monitoring (SHM) systems are integrated with a variety of damage detection methods, which are global and local in nature. Limitations in local methods necessitate the non-destructive and global techniques for damage diagnosis leading to continuous development in vibration based damage detection (VBDD) methods. In this context, this paper develops and applies mode shape component specific damage indices namely; vertical damage index (DIV) and lateral damage index (DIL) based on the Modal Flexibility (MF) concept to detect and locate damage in the main cables and hangers of a suspension bridge. The application of these damage indices is illustrated under single, multiple and complex damage scenarios. Effects of noise in the modal data and higher order vibration modes on the damage detection capability are also captured. Results confirm the applicability of the proposed vertical damage index DIV to accurately detect damage in real suspension bridges using only the first few modes. Bridge engineers can use the proposed technique to monitor the health of suspension bridges and prevent their unexpected failure.

Fisher information-based optimal input locations for modal identification

A method is proposed to determine optimal locations of input forces for experimental modal analysis. The input locations are optimal in a sense that the measured vibration responses, under inputs thus located, contain maximum information about unknown modal parameters of interest. These optimal input locations are obtained by maximizing the trace of the associated Fisher Information Matrix (FIM). Analytical expressions are obtained for the different derivatives involved, by expressing acceleration responses in terms of pseudo-modal responses. A dimensionless scaling is introduced to normalize the effect of different types of modal parameters. An explicit relation is also derived between the FIM and the mode shape components at input locations. It is shown that, in certain experimental scenarios, this relation may be used to directly obtain the optimal input locations from only the mode shapes. An extensive series of numerical simulations, including situations of single/multiple inputs and single/multiple modes of interest, is used to illustrate the proposed approach. It is observed that the modal parameters, identified using any suitable modal identification technique, have the lowest estimation uncertainty when the inputs are optimally located. As an application in damage detection, it is shown that modal parameters estimated from experiments with non-optimally located inputs may lead to incorrect damage localization. The proposed method is also applied to data from experiments performed on a laboratory scale truss model.

Horizontal Resonance of a 13 Story Building Under External Machine Vibrations

Excessive vibrations seriously affected the comfort of residents living on the upper floors of a high-rise shear walled building in Beijing. The ambient vibration tests were conducted to measure the floor acceleration responses, which were found to contain almost periodic signals likely to be excited by vibration sources with frequency of about 1.5[Formula: see text]Hz. The transverse vibration levels of the building above the 8th floor are not acceptable as revealed by the one-third octave spectra and weighted acceleration levels according to the ‘Standard for Allowable Vibration of Building Engineering’ of China. The modal properties of the building are identified by a Bayesian FFT method, indicating that the resonance between the building and the vibration sources caused the excessive vibrations. For comparison, the vibration test of an adjacent building with the same structural design was also conducted, together with modal analysis by the finite element method. It is found that as the story level increases, different trends of amplification in floor root mean square (RMS) acceleration and mode shape component of the two buildings cause different vibration levels. After tests outside the residence community, the main vibration sources were identified to be the working machines in two stone processing factories a few hundred meters away from the building. The vibration tests with measurements in the building and near the vibration sources with different number of machines in the two factories were also conducted. The results show that the vibration levels of the building can be controlled below the acceptance value by reducing the number of machines.

Influence of Modal Mass Participation in Damage Detection of Cable Structures

Increasing applications of large diameter and long-span cables as key structural components in cable-supported structures are becoming evident. However, these cables accumulate damage over time during their life cycle and such damage needs to be detected to avoid the detrimental influences on the serviceability and ultimate capacity of the structure. In this context, vibration-based damage detection (VBDD) methods have been used in some structures. Cable structures however exhibit complex vibration patterns, namely with vertical, lateral, torsional and coupled modes which complicate the VBDD procedure. To address this matter, this paper proposes a new approach for detecting and locating damage in cables using component-specificdamage indices (DIs) based on the modal flexibility method considering the modal mass participation. Application of this new procedure is illustrated through two case studies: (a) a suspended cable and (b) a three dimensional (3D) suspension bridge structure. Results verify that the DIs based on the lateral and vertical components of mode shapes, identified through their modal mass participation factors, are effective for detecting and locating damage in the suspended cables and main cables in a suspension bridge respectively, under a range of damage scenarios. The research outcomes of this paper confirm that the modal mass participation factor is an important signature in the damage detection of a structure using VBDD techniques.

Synchronization of dynamic response measurements for the purpose of structural health monitoring

This paper presents a technique for offline time synchronization of data acquisition systems for linear structures with proportional damping. The technique can be applied when direct synchronization of data acquisition systems is impossible or not sufficiently accurate. The synchronization is based on the acquired dynamic response of the structure only, and does not require the acquisition of a shared sensor signal or a trigger signal. The time delay is identified from the spurious phase shift of the mode shape components that are obtained from system identification. A demonstration for a laboratory experiment on a cantilever steel beam shows that the proposed methodology can be used for accurate time synchronization, resulting in a significant improvement of the accuracy of the identified mode shapes.

A two-step method composed of wavelet transform and model updating method for multiple damage diagnosis in beams

In the present study, a two-step approach comprised of wavelet transform and model updating method for multiple structural damage localization and quantification in beams is proposed. The first step is commenced with applying wavelet transform to axial components of mode shapes so as to predict where the damages are located. During this step, wavelet transform might mistakenly detect a number of elements as impaired due to both sampling interval and middle support effect. In the next step, by defining a damage sensitive objective function consisting of natural frequencies and mode shapes, damage intensities at predicted locations will be computed via model updating method employing ECBO (Enhanced Colliding Bodies Optimization) algorithm. The problem with mistakenly predicted damaged locations will be addressed by reported damage intensities during the second step. The present study also indicates that this two-step method greatly assists in reducing the number of variables during model updating process leading to more precise results. Three numerical examples with multiple damages and noisy modal data are studied in order to guarantee the efficacy of the method. Moreover, one numerical example is solved with a number of other Meta-Heuristic algorithms including GA (Genetic Algorithm), CSS (Charged System Search), Pattern Search and Cuckoo algorithm whose results are compared to ECBO algorithm with the intention of ensuring the veracity of ECBO results. The results vividly demonstrate that this method is highly efficient even in the presence of noise.

Vibration characteristics and damage detection in a suspension bridge

Suspension bridges are flexible and vibration sensitive structures that exhibit complex and multi-modal vibration. Due to this, the usual vibration based methods could face a challenge when used for damage detection in these structures. This paper develops and applies a mode shape component specific damage index (DI) to detect and locate damage in a suspension bridge with pre-tensioned cables. This is important as suspension bridges are large structures and damage in them during their long service lives could easily go un-noticed. The capability of the proposed vibration based DI is demonstrated through its application to detect and locate single and multiple damages with varied locations and severity in the cables of the suspension bridge. The outcome of this research will enhance the safety and performance of these bridges which play an important role in the transport network.

Experimental characterisation of railway wheel squeal occurring in large-radius curves

Tonal squeal noise (i.e. the high-amplitude singing of a railway wheel with pure tone components) is emitted by some trailing inner wagon wheels on heavy haul trains traversing 1000 m radius curves on the iron ore export line in South Africa. Field measurements have shown that the trailing inner wheels that squeal are subject to predominantly longitudinal creepage with little-to-no lateral creepage. The longitudinal creepage acting at the contact of the squealing wheels exceeds 1%, which supports the likelihood of creep saturation and subsequent squealing due to unsteady longitudinal creepage in the large-radius curves. Experimental modal analysis of the wheel types identified to be relevant to squeal has revealed that for each unstable frequency, two eigenmodes are likely to be important: one that has a large mode shape component at the wheel–rail contact in the circumferential direction and another that has a large mode shape component at the wheel–rail contact in the radial direction. A frictional self-excitation mechanism based on mode-coupling is favoured as being responsible for squeal excited in large-radius curves.

デッキプレートスラブの固有振動解析と平板への換算に関する検討

The natural frequencies and the mode shapes of deck plate slabs in steel structure buildings were measured. An eigenvalue analysis was carried out on the measured slab by the finite element method with coupling of in-plane and bending displacement, and the result showed good correspondence with the measurement. By using the eigenvalue analysis by the finite element method, conversion of deck plate slabs to flat plates were studied, and relation between ratio of side length in mode shape component and converted plate thickness was shown. The conversion method was verified in the vibration analysis of the actual building floor.

A two-stage fast Bayesian spectral density approach for ambient modal analysis. Part I: Posterior most probable value and uncertainty

A two-stage fast Bayesian spectral density approach is proposed in this study to extract modal properties for the cases of separated modes and closely spaced modes. A novel technique for variable separation is developed so that the interaction between spectrum variables (i.e., frequency, damping ratio as well as the spectral density of modal excitation and prediction error) and spatial variables (i.e., mode shape components) can be decoupled completely for both cases. In a first stage, the spectrum variables can be identified through a so-called ‘fast Bayesian spectral trace approach’ (FBSTA) by employing the statistical properties of the sum of auto-spectral density, while the spatial variables can be estimated in a follow up second stage through ‘fast Bayesian spectral density approach’ (FBSDA) by using the statistical information of the entire spectral density matrix. This study also reveals the intrinsic relationship between FBSDA and ‘fast Bayesian FFT approach’ formulated recently when multiple sets of measurements are available. The newly developed two-stage Bayesian approach allows for a fast computation of the most probable values and covariance matrix of modal properties. The companion paper is devoted to assembling the local mode shape components corresponding to different setups to form the overall mode shapes using a Bayesian statistical framework and verifying the proposed algorithms through simulated and field testing data.

A novel Bayesian approach for structural model updating utilizing statistical modal information from multiple setups

In this paper, a fast Bayesian methodology is presented for structural model updating utilizing modal information from multiple setups. A two-stage fast Bayesian spectral density approach formulated recently is firstly employed to identify the most probable modal properties as well as their uncertainties. The model updating problem is then formulated as one minimizing an objective function, which can incorporate statistical information about local mode shape components corresponding to different setups automatically, without prior assembling or processing. A fast analytic-iterative scheme is proposed to efficiently compute the optimal parameters so as to resolve the computational burden required for optimizing the objective function numerically. The posterior uncertainty of the model parameters can also be derived analytically and the computational difficulty in estimating the inverse of the high dimensional Hessian matrix required for specifying the covariance matrix is also properly tackled. The efficiency and accuracy of all these methodologies are verified by numerical examples.

Modal parameter based structural identification using input–output data: Minimal instrumentation and global identifiability issues

It is of interest to the modal testing and structural health monitoring community to be able to identify the mass and stiffness parameters of a system from its vibration response measurements. On the other hand, incomplete instrumentation of the monitored system results in measured mode shapes which are incomplete and may lead to non-unique identification results. In this study, the problem of mass normalized mode shape expansion, and subsequent physical parameter identification, for shear-type structural systems with input–output measurements is investigated. While developing a mode shape expansion algorithm, the issue of global identifiability of the system is also addressed vis-à-vis instrumentation set-ups. Several possible minimal and near-minimal instrumentation set-ups which guarantee a unique estimation of the unmeasured mode shape components from the measured components are identified for various experimental designs. An input–output balance approach, applicable to any general structural model, is proposed to mass normalize the mode shape components observed at the instrumented degrees of freedom. Using the proposed mode shape expansion and the input–output balance procedures, along with the modal orthogonality relations, the mass and stiffness matrices of the system can be estimated. The advantage of the algorithm lies in its ability to obtain a reliably accurate identification using the minimal necessary instrumentation with no a priori mass or stiffness information. The performance of the proposed algorithm is finally discussed through numerical simulations of forced vibration experiments on a 7 degree of freedom shear-type system.

Quadratic Mode Shape Components From Ground Vibration Testing

Points on a vibrating structure generally move along curved paths rather than straight lines. For example, the tip of a cantilever beam vibrating in a bending mode experiences axial displacement as well as transverse displacement. The axial displacement is governed by the inextensibility of the neutral axis of the beam and is proportional to the square of the transverse displacement; hence the name “quadratic mode shape component.” Quadratic mode shape components are largely ignored in modal analysis, but there are some applications in the field of modal-basis structural analysis where the curved path of motion cannot be ignored. Examples include vibrations of rotating structures and buckling. Methods employing finite element analysis have been developed to calculate quadratic mode shape components. Ground vibration testing typically only yields the linear mode shape components. This paper explores the possibility of measuring the quadratic mode shape components in a sine-dwell ground vibration test. This is purely an additional measurement and does not affect the measured linear mode shape components or the modal parameters, i.e., modal mass, frequency, and damping ratio. The accelerometer output was modeled in detail taking into account its linear acceleration, its rotation, and gravitational acceleration. The response was correlated with the Fourier series representation of the output signal. The result was a simple expression for the quadratic mode shape component. The method was tested on a simple test piece and satisfactory results were obtained. The method requires that the accelerometers measure down to steady state and that up to the second Fourier coefficients of the output signals are calculated. The proposed method for measuring quadratic mode shape components in a sine-dwell ground vibration test seems feasible. One drawback of the method is that it is based on the measurement and processing of second harmonics in the acceleration signals and is therefore sensitive to any form of structural nonlinearity that may also cause higher harmonics in the acceleration signals. Another drawback is that only the quadratic components of individual modes can be measured, whereas coupled quadratic terms are generally also required to fully describe the motion of a point on a vibrating structure.