Mode Shape Vectors Research Articles

Multi-Objective Model Updating of Tower Structures Featuring Closely Spaced Modes

In this contribution, we present a multi-objective model updating approach for the purpose of damage localisation and quantification on a girder mast structure based on closely spaced modes. In structural health monitoring (SHM), modal quantities, such as eigenfrequencies and mode shapes, serve as monitoring parameters. Specifically, model updating strategies adjust the local structural stiffness of a simulated structure to reduce the difference between the simulation results and the experimentally identified modal quantities. The resulting structural stiffness represents the damaged state of the structure, thus providing valuable information about the occurred damage. Symmetrical tower structures are challenging for model updating approaches due to their closely spaced bending modes, which are prone to uncertain alignments between the identified mode shapes and the model mode shapes. As a result, the naive comparison between the mode shapes of the real structure and the model often yields poor results. In order to overcome this problem, we adapt the concept of the subspace of order 2 modal assurance criterion (S2MAC), i.e., finding the best fit to a mode shape vector in a subspace spanned by two bending modes, hence enabling a meaningful comparison of the changes in the dominant mode shapes. The proposed method is validated using an experimental 9 m tall girder mast structure featuring reversible damage mechanisms. Mode shapes and eigenfrequencies, identified using the Bayesian operational modal analysis (BayOMA) method, are employed for multi-objective model updating, allowing for the estimation of several Pareto-optimal solutions. The structure is parameterised with two damage parameters, accounting for the location and severity, which limits the optimisation to only one damaged beam at any given time. The damage localisation is performed for data sets recorded over a time span of several months, including three different damage positions, resulting in a successful validation of the proposed

Dynamic modeling and experimental verification of an L-shaped pipeline in aero-engine subjected to base harmonic and random excitations

The aero-engine pipeline is inevitably subjected to base harmonic and random excitations during operation, but the vibration response analysis of the aero-engine pipeline under base random excitation has been rarely incorporated in the open research. Therefore, a dynamic model of an L-shaped pipeline combining the transfer matrix method (TMM) and lumped parameter method is proposed to solve the vibration response of the aero-engine pipeline under base harmonic and random excitations, and the natural and response characteristics of the pipeline are investigated by simulation and experiment. Firstly, the transfer matrix of the L-shaped pipeline is derived, and the natural frequencies and the corresponding mode shape vectors are obtained. Then, combined with the lumped parameter method, the transfer matrix of the L-shaped pipeline under base harmonic excitation is reconstructed and its frequency domain response is solved. On this basis, the virtual excitation method is introduced to address the dynamic responses of pipelines under base random excitation. Finally, the vibration test system for the L-shaped pipeline subjected to base excitation is built, and the vibration response solution method is verified through the comparison of simulation and experimental results. The work can provide an effective solution for the vibration response analysis of aero-engine pipelines under base harmonic and random excitations.

Dynamic performance of sleeper-damping track with elastic side-supporting pad

ABSTRACT Vibration and noise induced by urban rail transit has been a growing public concern worldwide. A new sleeper-damping track with elastic side-supporting pad (SDTESSP) has been proposed to reduce the environmental vibration caused by metro trains. To evaluate the dynamic performance of SDTESSP system, a refined 3D vehicle-SDTESSP coupled dynamics model was developed in this work. Firstly, the SDTESSP model was established based on the mode superposition method, and the mode shape vectors, and natural frequencies are obtained through a finite element method, and the finite element model of the SDTESSP was verified by the mechanical property and impact tests. Then, the influences of rail irregularities, vehicle running speed, and structural parameters of elastic side-supporting pad (ESSP) on the dynamic performances of the vehicle and SDTESSP are evaluated in detail. Finally, under the same track stiffness, the dynamic responses of the vehicle and track system caused by the conventional elastic under-supporting pad (USP) and the new side-supporting pad (SSP) are compared. The results show that the proposed dynamics model can avoid the difficulty in casting dynamics equation and enables detailed evaluations for the dynamic performance of the SDTESSP system. Furthermore, the optimal thickness and taper range of the ESSP are presented by comprehensively considering all dynamic indexes, and the damping performance of the sleeper-damping track using SSP is better than that using USP.

A Bayesian approach to global mode shape identification using modal assurance criterion-based discrepancy model

Modal identification of large-scale engineering structures may require multiple measurements obtained from different locations, due to the limitations in the instrumentation equipment. In such cases, the global values for modal frequencies and damping ratios can be estimated over the measurement setups by simply obtaining their statistical expected values. Estimation of global mode shape vectors, however, arises as a much more challenging problem since the local mode shapes confine at different degrees of freedom of the measured structure. To overcome this problem, a novel Bayesian methodology is presented in this study using a modal assurance criterion-based discrepancy model. By making use of this model, a novel solution procedure is presented to the existing literature, and mode shape normalization-based problems arising in the estimation of covariance matrix are solved. The computational efficiency of the proposed method is verified by two numerical and one real data example. The conducted investigations reveal that the presented methodology shows a good performance in terms of computational accuracy.

Development of a Novel Dynamic Modeling Approach for a Three-Axis Machine Tool in Mechatronic Integration

This paper proposes a novel, fast, and automatic modeling method to build a virtual model with minimum degrees of freedom (DOFs) without the need for FE models or human judgment. The proposed program uses the iterative closest point (ICP) algorithm to analyze the mode shape vector of structural dynamic characteristics to define the position and DOFs of the joints between structural components. After the multi-body dynamics model was developed in software, it was converted into an SSM to connect the servo loop model. Then, the mechatronic integration analysis was performed to verify the dynamic characteristics of the tool center point (TCP) and the workbench in the experiment and simulation. The model created by the proposed identification process has a small DOF and can accurately simulate the dynamic characteristics of a machine. This model can be used for dynamic testing and control strategy development in mechatronic integration.

Experimental assessment of dynamic and static based stiffness indices for RC structures

This paper presents the results obtained from experimental modal analysis and static load testing to examine the suitability of the dynamic parameters to estimate stiffness deterioration severity in RC structures. The study investigates the accuracy of different dynamic-based stiffness indices compared to static-based flexural and shear stiffness indices and how different damage levels and scenarios impact them. The flexural damage scenario was found to have the weakest effect on stiffness deterioration, indicated by the frequency-based stiffness index, while shear damage at quarter-span has the greatest influence. The frequency-based stiffness index indicates a stiffness increase when the damage level reaches 0.15, which is believed to be related to the composite effect in RC structures, as discussed by Abdul Razak and Fayyadh (2013). The combined modal parameters stiffness index revealed that the flexural damage scenario has the weakest influence on stiffness deterioration, while shear damage closer to the support has the highest effect on stiffness deterioration. The main difference between the combined parameters index and the other two dynamic indices is that the combined parameters index detects global stiffness changes through frequency, while the local stiffness changes are detected through the mode shape vectors. The general trend of the dynamic-based indices shows that the MAC-based stiffness index gives the lowest sensitivity for stiffness change for all damage levels, while the combined parameters-based stiffness index gives the highest sensitivity among the dynamic parameters indices to detect the stiffness deterioration for all damage levels. The static-based flexural stiffness index shows lower sensitivity to stiffness deterioration than the static-based shear stiffness index. The static-based stiffness indices show better sensitivity than the dynamic-based indices; however, the static load test is time-consuming, equipment and labour intensive, and causes significant disruptions to the existing use of the structures. Thus, the dynamic parameters-based stiffness indices are more suitable for existing structures.

Performance evaluation of piezo sensors with respect to accelerometers for 3D modal analysis of structures

Strain modal analysis, as a new domain in the health monitoring field, needs to be studied in depth for experimental modal testing. Towards this purpose, this paper experimentally investigates the efficacy of piezo sensors for structural identification for structural health monitoring under different excitations applicable to large-scale structures. The piezo sensors are evaluated against industry-standard accelerometers by experimental modal testing of a scaled-down model of a pedestrian foot over bridge. The model is excited under the impact hammer, electro-dynamic shaker-based sweep and random excitations, and pedestrian motion (PM)-based low-amplitude excitations. Piezo sensors are found to be capable of capturing the modal parameters (modal frequencies, damping ratios and mode shape vector) under all the excitations with excellent correlation with respect to accelerometer-based parameters. However, some modes are missed under the shaker and PM-based excitations compared to the impact hammer-based excitations for both accelerometers and piezo sensors. Modal parameters of lower modes are successfully extracted under low-level pedestrian excitations, the most efficient type of excitation acting in operational conditions. High modal assurance criteria values between the strain and the displacement mode shapes establish the piezo sensors as effective for strain-based vibration testing and structural identification.

A Gaussian process regression‐based surrogate model of the varying workpiece dynamics for chatter prediction in milling of thin‐walled structures

AbstractSince the dynamics of thin‐walled structures instantaneously varies during the milling process, accurate and efficient prediction of the in‐process workpiece (IPW) dynamics is critical for the prediction of chatter stability of milling of thin‐walled structures. This article presents a surrogate model of the IPW dynamics of thin‐walled structures by combining Gaussian process regression (GPR) with proper orthogonal decomposition (POD) when IPW dynamics at a large number of cutting positions has to be predicted. The GPR method is used to learn the mapping between a set of the known IPW dynamics and the corresponding cutting positions. POD is used to reduce the order of the matrix assembled by the mode shape vectors at different cutting positions, before the GPR model of the IPW mode shape is established. The computation time of the proposed model is mainly composed of the time taken for predicting a known set of IPW dynamics and the time taken for training GPR models. Simulation shows that the proposed model requires less computation time. Moreover, the accuracy of the proposed model is comparable to that of the existing methods. Comparison between the predicted stability lobe diagram and the experimental results shows that IPW dynamics predicted by the proposed model is accurate enough for predicting the stability of milling of thin‐walled structures.

Vibration Measurement of a Metal Sheet Using Single-Camera Digital Image Correlation with Projection Components

Digital image correlation has emerged as a popular method for the dynamic performance measurement of metallic and polymer sheets, owing to the benefits of being a noncontact, full-field, and high-precision method. Two or more high-speed cameras are required for full-field vibration measurements with three-dimensional digital image correlation, which is generally costly. Perpendicular view to the specimen surface is conventional in two-dimensional digital image correlation, and the out-of-plane displacement is regarded as a part of systematic errors. In this study, a single view method was implemented with no complex optical settings. The full-field vibration displacement of the metal sheet was measured with projection components, and the first four orders of displacement modes were identified. Finite element analysis and traditional experimental modal analysis were then implemented to validate the effectiveness and accuracy of the proposed approach. The results show that the dynamic parameters, including the natural frequencies and mode shapes, were well consistent. Meanwhile, there is a significant difference in the length of mode shape vectors. The number of measurement points in the proposed method is 2016, which is far more than the number of measurement points in the traditional experimental modal analysis. This would be convenient and beneficial for damage identification towards thin-wall parts including turbine blade with the continuum hypothesis of mode shapes and a single-camera DIC system. It is worth noting that this is effective with conditions of small deformation vibration and no rigid-body rotation.

Mode Tracking of Unidirectional Carbon-Based Composite Structures Using Modified Mode Shape Vectors

A comparison of mode shapes in isotropic structures can be efficiently performed using the modal assurance criterion (MAC) to determine the similarity between mode shape vectors. However, the unidirectional, carbon-based composite (UCBC) structure shows different dynamic characteristics according to the carbon fiber orientation, even for the same structural configuration. The MAC of a certain mode may result in a poor value for the CBC structures in the case of the existence of the distorted mode shape vector from reinforced carbon fibers. In this study, the mode tracking of the UCBC structure is proposed using the MAC value only under the modified mode shape vector to enhance the MAC value between relevant modes. Because the mode shape vectors of the UCBC structure are altered from those of the isotropic structure owing to the reinforced stiffness along the carbon fiber orientation, the modified mode shape vectors are calculated by multiplying the original vectors with the proposed modification window. The proposed method was verified for simple UCBC structures with five different carbon fiber orientations, from 0° to 90°. The UCBC structures were tracked for five modes, three bending and two torsional, and the results were discussed with reference to earlier study results.

Damage Identification for Shear-Type Structures Using the Change of Generalized Shear Energy

Structural damage identification has become an important topic in the field of civil engineering in recent years. The shear-type structure, such as shear frame structure, is a common type used in civil engineering. In this paper, a damage identification method based on the change of generalized shear energy is proposed for shear-type structures. The main steps of the proposed method are as follows. Firstly, the element stiffness matrix in the structural finite element model is decomposed to obtain the elementary shear force vector. Secondly, the elementary generalized shear energy is calculated by the dot product of the vibration mode shape vector and the elementary shear force vector. Thirdly, structural damage locations can be determined by the changes of elementary generalized shear energy. Finally, more accurate damage localization and quantification are achieved by solving the mode shape sensitivity equation. A 20-storey numerical example and a three-storey experimental model are used to demonstrate the proposed damage identification algorithm. From the numerical and experimental results, it was found that the proposed approach can accurately identify the location and extent of the damage in the shear structures even if the data contain noise. It has been shown that the presented algorithm may be useful in the damage identification of shear-type structures.

Identification of Mode Shapes of a Composite Cylinder Using Convolutional Neural Networks.

The aim of the following paper is to discuss a newly developed approach for the identification of vibration mode shapes of multilayer composite structures. To overcome the limitations of the approaches based on image analysis (two-dimensional structures, high spatial resolution of mode shapes description), convolutional neural networks (CNNs) are applied to create a three-dimensional mode shapes identification algorithm with a significantly reduced number of mode shape vector coordinates. The CNN-based procedure is accurate, effective, and robust to noisy input data. The appearance of local damage is not an obstacle. The change of the material and the occurrence of local material degradation do not affect the accuracy of the method. Moreover, the application of the proposed identification method allows identifying the material degradation occurrence.

Robust multi-dataset identification with frequency domain decomposition

Right after its invention in the late nineties, the Frequency Domain Decomposition (FDD) identification technique became very popular in the operational modal analysis community due to its simplicity and robustness. The underlying idea of this technique consists of computing the singular value decomposition of the Power Spectral Densities (PSDs) estimated from the measured vibration responses with the periodogram (also known as “Welch’s”) approach to identify the natural frequencies and mode shape vectors of the tested structural system. When dealing with multi-dataset output-only modal analysis, the classic approach for extracting the global mode shape vectors from all the measured datasets consists of estimating the mode shape parts corresponding to each dataset, and then scaling the different parts with the aid of the reference sensors. In this paper, two new merging approaches are proposed to scale the different mode shape parts. The first consists of (i) re-scaling the different PSDs prior to the identification; (ii) forming a global matrix containing all the re-scaled PSDs; and (iii) applying the FDD approach to the global PSD matrix to estimate the global mode shape vectors. The second merging strategy consists of (i) forming a global matrix containing all the PSDs without any prior re-scaling; (ii) applying the FDD approach to the PSD matrix to estimate the global mode shape vectors; and (iii) re-scaling the different mode shape parts by making use of the reference singular vectors. In order to illustrate the benefits of the two proposed merging approaches with regards to their classic counterpart from a practical perspective, a real-live application example is presented as the last part of the paper.

Dynamic performance evaluation of rail fastening system based on a refined vehicle-track coupled dynamics model

Damage of rail fastening system often occurs due to severe vehicle-track dynamic interactions. To evaluate the dynamic performance of the rail fastening system subject to dynamic interactions between vehicle and track, a refined vehicle-track coupled dynamics model involving an elaborated model of rail fastening clip (RFC) is developed in this work. Firstly, the RFC model is established based on the mode superposition method by which the difficulty in deducing dynamics equation due to the complex shape of RFC can be easily resolved. Then, the mode shape vectors and natural frequencies of RFC which required by the mode superposition method are obtained through a numerical modal analysis, and verified by a modal test. Finally, through the contact relationship between RFC and rail, the RFC model is implemented into the vehicle-track coupled dynamics model. The reliability of the proposed model is supported by a field test data, and the dynamic performances of rail fastening systems along the track subject to vehicle-track dynamic interactions can be efficiently and effectively evaluated. By applying the proposed model, the influences of rail corrugations, vehicle speeds, and rail pad stiffness on dynamic performances of the rail fastening system are evaluated in detail.

Element damage assessment in semi rigid connected structures using modal domain data

Current vibration-based damage assessment techniques assumes that the joints in a structure behave either perfectly rigid or perfectly flexible. However, joints in the bolted structures are semi-rigid, which means the elements may show rotations to a certain degree on the basis of a fixity factor. In this regard, the present work proposes an alternate solution to conventional inverse damage identification problems by providing due weightage to the joint rigidity. The beam-column joints for this study are made semi-rigid by reducing their end fixity factor. An inverse problem is formulated here using changes in the natural frequencies and mode shape vectors caused due to damages, which is then solved by a unified particle swarm optimisation algorithm. The efficacy of this procedure is demonstrated by conducting a few numerical and experimental studies. The outcomes suggest that the developed procedure is able to locate and quantify damages in a semi-rigidly connected structure with significant accuracy.

Modal Identification of damped vibrating systems by iterative smooth orthogonal decomposition method

The smooth orthogonal decomposition method (SOD) is an efficient algorithm that can be used to extract modal matrix and frequencies of lightly damped vibrating systems. It uses the covariance matrices of output-only displacement and velocity responses to form a generalized eigenvalues problem (EVP). The mode shape vectors are estimated by the eigenvectors of the EVP. It is stated in this work that the accuracy of the SOD method is mainly affected by the correlation characteristic of modal coordinate responses. For the damped vibration systems, biases will be contained in the results of using the SOD. Therefore, an iterative smooth orthogonal decomposition (ISOD) method is proposed to identify modal parameters of the damped system from the covariance matrices of the displacement, velocity, and acceleration responses. The modal matrix given by the SOD method is updated in each iteration step using a transformation matrix. The transformation matrix can be efficiently computed using a set of analytical formulations. Meanwhile, natural frequencies and damping ratios are obtained by using a simple search method. The performance of the proposed ISOD method is verified by numerical and experimental studies. The results demonstrate that, by considering the correlation of modal responses, the ISOD method can be used to extract accurately the modal information of vibration systems with coupled modes.

Blind modal estimation using smoothed pseudo Wigner–Ville distribution and density peaks clustering

The determination of the number of active modes is always an obstacle to the identification of dynamic systems. To tackle this issue, a new blind modal estimation method is proposed through which the number of active modes does not need to be presupposed, even in underdetermined blind source separation cases. Firstly, the spatial time–frequency distribution matrices are constituted from the smoothed pseudo Wigner–Ville distribution of acceleration signals. The auto-source time–frequency points that contain modal information about only one mode are selected and then the density peaks clustering algorithm is introduced to identify the mode shape vectors. Finally, one can reconstruct the time–frequency distributions of modal coordinates and estimate the natural frequencies and damping ratios. The method is verified by a numerical study and a vibration experiment. The anti-noise study demonstrates that the method performs well.

Representation of wind turbine blade responses in power production load cases by linear mode shapes

AbstractThe wind turbine industry is designing large MW size turbines with very long blades, which exhibit large deflections during their operational life. These large deflections decrease the accuracy of linear models such as linear finite element and modal‐based models, in which the structure is represented by linear mode shapes. The aim of this study is to investigate the competence of the mode shapes to represent the large blade responses in normal operation load cases. For this purpose, blade deflections are projected onto the linear modal space, swept by mode shape vectors. The projection shows the contribution of each mode and the projection error. The blade deflections are calculated by a nonlinear aero‐servo‐elastic solver for power production fatigue load cases with normal turbulence. The mode shapes are calculated at the steady‐state deflected blade position computed at different wind speeds. Three reference turbine blades are used in the study to evaluate the effects of various blade design parameters such as length, stiffness, mass, and prebend. The results show that although the linear mode shapes can represent the flapwise and edgewise deflections accurately, axial and torsional deflections cannot be captured with good accuracy. The geometric nonlinear effects are more apparent in the latter directions. The results indicate that the blade deflections occur beyond the linear assumptions.

Damage Detection for Large-Scale Grid Structure Based on Virtual Axial Strain

To identify the damaged beams in large-scale spatial structure, a damage indicator based on virtual axial strain calculated from mode shape vectors was proposed. The damage detection process was performed based on the dynamic simulation flowchart. Firstly, random signals were used for excitation and the damage was simulated by decreasing beam elasticity modulus. Then, the NEWMARK-β precision direct integral method was appreciated for calculating time history response. Finally, the frequency-domain decomposition method only using output response signal was selected for modal parameter estimation. A double-layer grid structure was taken as example for verifying the damage detection method. Results indicate that the proposed indicator was insensitive to environmental noise and capable of localizing multiple damaged members in space structure without the baseline data.

Feasibility study on model-based damage detection in shear frames using pseudo modal strain energy

This paper proposes a model-based approach for structural damage identification and quantification. Using pseudo modal strain energy and mode shape vectors, a damage-sensitive objective function is introduced which is suitable for damage estimation and quantification in shear frames. Whale optimization algorithm (WOA) is used to solve the problem and report the optimal solution as damage detection results. To illustrate the capability of the proposed method, a numerical example of a shear frame under different damage patterns is studied in both ideal and noisy cases. Furthermore, the performance of the WOA is compared with particle swarm optimization algorithm, as one the widely-used optimization techniques. The applicability of the method is also experimentally investigated by studying a six-story shear frame tested on a shake table. Based on the obtained results, the proposed method is able to assess the health of the shear building structures with high level of accuracy.