A novel mode shape identification approach for structures having planes with rigid-like behavior

Cyclostationarity and the cepstrum for operational modal analysis of MIMO systems—Part II: Obtaining scaled mode shapes through finite element model updating

The design and assessment of structural dynamics behaviour often rely on the ‘modal properties’ (primarily modal frequencies, damping ratios and mode shapes) of a structure. Modal properties are frequently identified in vibration related applications as their information can benefit design theory validation and in-situ property updating and monitoring. Given the prospect of flexibility and high implementation economy, operational modal analysis (OMA) aims to identify the modal properties using only ambient vibration response whose amplitude can be small. When instrument noise becomes a non-negligible part of measured data, the identified result can have significant uncertainty, calling for proper uncertainty quantification. Amidst the complex composition of ambient vibration data of civil structures, conventional noise assumption may not always apply. Motivated by issues about uncertainty modelling and computation, this thesis contributes to the theory and computational algorithm of OMA from a Bayesian perspective in the frequency domain. Focusing on the resonance band of vibration modes, a Bayesian probabilistic approach provides a fundamental mathematical framework for OMA. The ‘posterior’ (i.e., given data) most probable value (MPV) of modal properties maximising the likelihood of data informs about the best estimate. The uncertainty associated with the estimate is quantified by the posterior covariance matrix, which can be calculated as the inverse of the Hessian of negative log-likelihood function (NLLF) at the MPV. Existing formulations assume independent and identically distributed (i.i.d.) prediction errors among different channels, which account for both instrument noise and modelling error. While the assumption significantly simplifies theory and computation, and may not matter when the signal-to-noise (s/n) ratio is high, it is prone to be violated in OMA. For example, large disparity among noise intensities may exist, and significant correlation can occur because of the presence of unmodelled dynamics, which may affect identification results. To understand and manage the consequence of these potential violations, a comprehensive investigation on the effect of prediction error models with different statistical assumptions is presented in this work. Beyond the i.i.d. assumption, two generalised prediction error models have been considered to account for the disparity and correlation that may arise in applications. Such investigation was previously hindered by computational issues because the general prediction error models feature more parameters carrying somewhat different mathematical roles from the existing ones, making the computation of posterior MPV and covariance matrix more challenging in the first place. As a contribution to computation and enabling scientific investigation, new efficient algorithms applicable to general prediction error models are developed in this work for their computation. As a contribution to theory, a mathematical analogy for general noise is established with i.i.d. model for OMA problems. Not only does this provide a simple approximate way for handling general noise with existing tools based on i.i.d. assumption, it also allows the understanding and experience accumulated with i.i.d. model to be generalised. As a spin-off of this opportunity, asymptotic formulae for identification uncertainty are developed for general prediction error models in this work, which yield a generalised definition for s/n ratio that provides a fundamental quantification of data quality in OMA. The theory and algorithms are developed and tested against synthetic, laboratory and field data. As a further contribution to computation, motivated by existing overheads and hurdles in computer code development, this work proposes an alternative semi-analytical means for calculating the Hessian of NLLF in OMA. The proposed method explores a mathematical identity in the theory of expectation-maximisation (EM) algorithm to allow the Hessian matrix to be calculated via Monte Carlo simulation, which offers an alternative means for counter-checking/benchmarking during computer code development. Theoretical implications of the identity are also explored, and numerical examples are presented to illustrate implementation aspects.

Methods for scaling mode shapes determined by operational modal analysis (OMA) have been extensively investigated in the last years. A recent addition to the range of methods for scaling OMA mode shapes is the so-called OMAH technique, which is based on exciting the structure by harmonic forces applied by an actuator. By applying harmonic forces in at least one degree-of-freedom (DOF), and measuring the response in at least one response DOF, while using at least as many frequencies as the number of mode shapes to be scaled, the mode shape scaling (modal mass) of all modes of interest may be determined. In previous publications on the method the authors have proven that the technique is easy and robust to apply to both small scale and large scale structures. Also, it has been shown that the technique is capable of scaling highly coupled modes by using an extended multiple reference formulation. The present paper summarizes the theory of the OMAH method and gives recommendations of how to implement the method for best results. It is pointed out, as has been shown in previous papers, that the accuracy of the mode scaling is increased by using more than one response DOF, and by selecting DOFs with high mode shape coefficients. To determine the harmonic force and responses, it is recommended to use the three-parameter sine fit method. It is shown that by using this method, the measurement time can be kept short by using high sampling frequency and bandpass filtering whereas spectrum based methods require long measurement times. This means that even for structures with low natural frequencies, the extra measurement time for scaling the mode shapes can be kept relatively short.

Asymptotic identification uncertainty of well-separated modes in operational modal analysis with multiple setups

This research investigates the dynamic behavior of a railway bridge using both experimental and numerical methods. Field tests were conducted to capture the bridge response to live loading conditions with acceleration data collected via uniaxial accelerometers placed at critical locations along the structure. The dynamic characteristics, including the natural frequencies and mode shapes, were determined using two analytical techniques: Fast Fourier Transform (FFT) and Operational Modal Analysis (OMA). While FFT provides a frequency domain analysis, OMA enables the estimation of modal parameters, such as natural frequencies, mode shapes, and damping ratios, using the bridge's response to operational forces. The results revealed that the fundamental frequency obtained from the OMA (2.163 Hz) was higher than that obtained from the FFT (1.95 Hz) and the Finite Element Analysis (FEA) model (1.65 Hz). Additionally, the OMA produced mode shapes that were closely aligned with those predicted by the FEA, validating the accuracy of the numerical model. This study highlights the advantages of OMA over FFT, particularly the ability to capture mode shapes, and underscores the importance of integrating OMA with FEA for a comprehensive dynamic assessment of bridge structures. These findings contribute to the growing body of knowledge on structural monitoring and provide practical insights into improving bridge safety and performance.

In order to derive mass normalized mode shapes from Operational Modal Analysis (OMA) techniques, additional experiments have to be conducted in the interest of scaling the determined mode shapes. Various investigations have been carried out based on the deterministic perturbed mass matrix, also known as the mass change method. However, the conventional form of this method requires a number of rather costly and sometimes impractical experiments on the structure. In this article, it is intended to use traffic as a stochastic source of mass change to calculate mass scaling factors. Adequate traffic stream on bridges can affect the Eigen properties of the structure efficiently. The vehicles on the bridge are only considered to affect the mass properties, and vehicle-bridge interaction is neglected. A simplified structural model of a bridge is considered and the traffic is modeled as lognormal distribution. Nodal masses induced by the vehicles are converted to time histories to avoid difficulties based on the moving mass problem. Consequently, a method is proposed to produce the modified mass matrix which can be used in the scaling factor equations. Finally, scaling factors of mode shapes are proposed by comparing the unperturbed structure and the perturbed mass matrix structure.

This research proposed and verified an innovative method to identify and locate structural damage using only the response of operational vibration, that is the displacement acquired by a non-contact optical method. The most efficient and economical way to detect damage within the structure is to monitor its structural health while in operation. However, the uncertainties and the randomness of ambient vibrations due to the operation and environments cause a challenge in conducting the operational analysis. Current technology limits the ability to collect data on the properties of the structure without the interruption of operation. Frequencies and mode shapes have been widely used in structural damage detection, but they are not sensitive enough and cannot provide sufficient information for identifying damage locations and their quantification. Therefore, the goal of this research is to design and verify a method to detect the damage, as well as its location and severity, of structures in operation without any physical contacts for data acquisition (i.e., non-contact based structural health monitoring (SHM)). Three algorithms are integrated into this SHM process. The first algorithm is the determination of structural characteristics (frequencies and mode shapes) of a vibrating structure from output-only data. Stochastic Subspace Identification (SSI) method is applied to measured displacements over time to extract the structural characteristics. The second algorithm is to estimate the scaling factor. The mode shapes obtained from the output-only model analysis are unscaled due to the absence of the information of input excitation forces. Mass Change Modal Scale (McMS) algorithm is used to estimate the modal scaling factors and determine the scaled mode shapes. The third algorithm is to estimate the structural system matrices (i.e., mass and stiffness matrices) and assess the damages. A Finite Element Model Updating (FEMU) is applied and the system matrices are updated from frequencies and scaled mode shapes. The damage within the structure can then be detected by analysing changes in mass and stiffness matrices. All three phases are verified by numerical simulation and laboratory experiments with deflections acquired by non-contact optical methods through video system. At last, to achieve the non-contact based SHM, a modal scaling method based on temperature change is proposed and verified by numerical simulation. Experimental program reveals that the proposed algorithm using McMS method is applicable to detect damage locations and their mass losses. With proposed non-contacted based SHM, the limitations of contact based sensor can be addressed, and the structural damage

Scaling-factor estimation using an optimized mass-change strategy

The vibrational effect is one of the primary considerations during construction due to the ambient excitation from the surrounding environment. Thus, it is necessary to find the most dependable Finite Element (FE) model for the actual structure to improve its quality and life span on site. In this context, the study included a 3-story building model that was analyzed using finite element analysis to determine its dynamic properties such as natural frequency and mode shape to avoid the resonance effect. It is due to the importance of obtaining a reliable and accurate model by verifying with the results of the Operational Modal Analysis (OMA). In general, the goal of this study is to update the FE model so that it is close to the actual scaled model based on the previous OMA study. As a result, an initial linear FE model of the 3-story building scaled model was created using CAD software based on actual geometry to describe the structure's physical properties. The FE model was then imported into CAE software, where the boundary condition and estimated material properties were assigned to determine the effect of random vibration. Through the pairing of Finite Element Analysis results and previous studies, 9 natural frequencies and 9 mode shapes were extracted. The Modal Assurance Criterion (MAC) was used to compare the mode shape of FE results against the OMA to determine the degree of consistency between paired mode shapes. A model updating process was carried out to reduce the discrepancy between the methods. The uncertainties arising from the initial conditions have been discussed in terms of the stiffness of the material used. The updated model allows for an evaluation of the structure's current actions as well as the development of models for a wide range of potential future research scenarios.

The paper presents how to identify natural frequencies and mode shapes of structures by Operational Modal Analysis (OMA) technique, in which the Frequency Domain Decomposition (FDD) method is used. This method is an experimental method only base on the data of measuring the dynamic response of the structures under the excitation due to ambient forces and operational loads to determine the vibration characteristics. Measure vibration (acceleration) and determine spectral density matrix, using the singular values decomposition method of spectral density matrix to determine the natural frequencies and mode shapes of structures. The calculation results show that the natural frequencies, the mode shapes form determined by the OMA technique is consistent with the calculation results according to the theory and show the reliability of the method.KeywordsNatural frequencyMode shapeIdentifyEMAOMAFDD

\nThis work presents a summary of the research study of operational modal analysis (OMA) application for condition monitoring of operating pipelines. Special focus is on the topicality of OMA for definition of the dynamic features of the pipeline (frequencies and mode shapes) in operation. The research was conducted using two operating laboratory models imitated a part of the operating pipeline. The results of finite-element modeling, identification of pipe natural modes and its modification under the influence of virtual failure are discussed. The work considers the results of experimental research of dynamic behavior of the operating pipe models using one of OMA techniques and comparing dynamic properties with the modeled data. The study results demonstrate sensitivity of modal shape parameters to modification of operating pipeline technical state. It is resumed about ability of operating pipeline condition monitoring by measuring dynamic deformations of the operating pipe and OMA techniques application for dynamic properties extraction.\n

In order to identify the modal parameters of time invariant three-dimensional engineering structures with damping and small nonlinearity, a novel isometric feature mapping (Isomap)-based three-dimensional operational modal analysis (OMA) method is proposed to extract nonlinear features in this paper. Using this Isomap-based OMA method, a low-dimensional embedding matrix is multiplied by a transformation matrix to obtain the original matrix. We find correspondence relationships between the low-dimensional embedding matrix and the modal coordinate response and between the transformation matrix and the modal shapes. From the low-dimensional embedding matrix, the natural frequencies can be determined using a Fourier transform and the damping ratios can be identified by the random decrement technique or natural excitation technique. The modal shapes can be estimated from the Moore–Penrose matrix inverse of the low-dimensional embedding matrix. We also discuss the effects of different parameters (i.e., number of neighbors and matrix assembly) on the results of modal parameter identification. The modal identification results from numerical simulations of the vibration response signals of a cylindrical shell under white noise excitation demonstrate that the proposed method can identify the modal shapes, natural frequencies, and ratios of three-dimensional structures in operational conditions only from the vibration response signals.

Safety assessment of structures for near-field blast-induced ground excitations using operational modal analysis

Smooth plant production is utmost expected by plant owners since any unplanned shutdown is very much undesirable as it may incur loss of company’s profits and other repercussions. Critical operational assets such as rotating machineries and their components, e.g., motor, pump, compressor, and turbines are crucial to be monitored in ensuring continuous operation of the plant. Therefore, vibration being one of the elements in predictive maintenance is much needed. This paper addressesthe monitoring and findings for predictive maintenance on high vibration of a make-up gas compressor in one of the oil & gas and petrochemical plant in Malaysia. The conventional vibration measurement using vibration analyzer was utilized to identify the severity of the machine vibrations through the overall vibration value obtained on the motor and compressor. A specialized vibration measurement technique, namely, Operational Modal Analysis (OMA) was further deployed to verify the presence of resonance on the equipment by obtaining its dynamic characteristics, i.e., natural frequency and mode shape. The findings showed clear indication of resonance occurrence as similar frequency (112 Hz) was identified from both vibration measurement and operational modal analysis.

Measuring configuration of multi-setup ambient vibration test