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Identified damping by Adhikari's method (−) and the proposed expression (−) for damper on fifth floor with varying magnitude. (a) Estimated componentˆCcomponentˆ componentˆC 55 of damping matrix associated with fifth floor and (b) the norm ||E|| of the relative error matrix.
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A sufficiently accurate mathematical representation of the viscous damping matrix from modal parameters is often limited to structures with light damping or an assumed structure of the damping matrix. These limitations are now circumvented by a novel expression, which reconstructs the damping matrix from the complex-valued eigenvectors and eigenval...
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... assessment of the identification accuracy is in Figure 3 small and without any increase for larger damper magnitudes, whereas it is initially larger and increases 296 monotonically for the benchmark method. individual laser optical displacement sensors, with a measurement range of 50 mm. In Figure 4 these sensors 321 are denoted 1 (top) to 5 (bottom), which are mounted on a common steel rack fixated to the strong ...
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... assessment of the identification accuracy is in Figure 3 small and without any increase for larger damper magnitudes, whereas it is initially larger and increases 296 monotonically for the benchmark method. individual laser optical displacement sensors, with a measurement range of 50 mm. ...
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Citations
... Such identified parameters are used in structural health monitoring and reliability analysis, among others. The identified modal parameters estimated using the developed techniques (Juang and Pappa 1985;Peeters and De Roeck 1999;Yang and Nagarajaiah 2013;Vicario et al. 2015;Rather and Bansal 2023) are widely used in damage detection (Moaveni et al. 2009;Bernal 2002;Balsamo et al. 2015;Moaveni et al. 2010), and model updating and physical parameters identification either with (Moaveni et al. 2009;De Angelis et al. 2002;Bajrić and Høgsberg 2018) or without (Moaveni et al. 2009;Yuen et al. 2006;Mukhopadhyay et al. 2015) a full set of sensors. Physical parameters can also be estimated directly from the measured data, circumventing the need for modal identification (Sun et al. 2013;Chatzi and Smyth 2009;Li et al. 2022). ...
The investigation of optimal sensor location problems in the field of vibration-based structural identification is predominantly performed for classically damped systems. In this study, a methodology is introduced to obtain the optimal sensor locations for identifying the structural (modal as well as physical) parameters of a nonclassically damped system. The optimal sensor locations are obtained by maximizing the determinant of the estimated Fisher information matrix out of all possible sensor combinations, such that the measurements would have maximum information about the parameters of interest. To deal with the inherently complex characteristics of the eigenvalues and mode shapes of nonclassically damped structures, the formulation is carried out in the state-space framework. To illustrate the approach numerically, primary-secondary systems exhibiting nonclassical damping behavior have been used. The methodology is also validated with experimental data from shake-table tests conducted on a primary-secondary structure. It is shown that the uncertainty in the estimated parameters is minimum when the sensors are placed at the optimal locations.
... Rayleigh's Viscous damping model, which is frequently used, has demonstrated the dissipation of energy in structural dynamics. In this framework, the dissipative forces exhibit a direct proportionality to the velocities of the system's degrees of freedom-DOF (Bajrić & Høgsberg, 2018). A pivotal consideration in structural dynamics is classifying the damping matrix as classical or non-classical. ...
This research delves into the consequences of neglecting modal coupling in non-classically damped systems, particularly within Rayleigh's lightly non-classical damping assumption. Damping devices are introduced into multi-degree-of-freedom frame systems to assess the impact of modal coupling on energy transfer in linear dynamic non-classical damping systems. Three distinct approaches for decoupling non-classically damped systems are introduced: the state-space approach for exact solutions, the lightly non-classical method for addressing the undamped complex eigenvalue problem, and the formulation of the quadratic eigenvalue problem to tackle the original eigenvalue problem. Modal coupling is evaluated using the modal assurance criterion. The study identifies instances of modal coupling in non-classically damped systems by analysing two scenarios. The proposed methodologies establish a robust foundation for exploring modal characteristics and advancing engineering solutions in various applications.
... In summary, previous studies typically assume classically damped systems. However, this contradicts the real-world scenario, where most structures do not adhere to classical damping conditions [39,40]. For example, structures like dams and controlled structures such as tuned mass dampers (TMD) do not preserve the orthogonality of modes [41,42]. ...
... The simulated modal data used in the present example consists of the MPVs of the modal frequencies, damping ratios, and partial mode shapes identified from measurements of the (10,20,30,40,50,60,70, and 80 DOFs from the base) along with their posterior uncertainties for the first four identified modes (N r 4). The modal data are usually obtained using the modal identification method. ...
This paper presents a Bayesian method for updating linear dynamical systems using complex modal data due to the effects of non-classical damping collected from multiple setups. The practical scenario of the availability of a limited number of sensors is avoided by considering measurements from multiple setups. Besides, the complex modal data is assumed to consist of the most probable values (MPVs) of the modal parameters and their posterior uncertainties, unlike the other approaches where only MPVs of the modal parameters are considered. System modal parameters are introduced as additional uncertain parameters to avoid the requirement of mode matching between the model-predicted and the measured modes. Since introducing the system mode shapes as additional parameters increases the problem dimensionality, the dynamic condensation technique is employed to reduce the finite element (FE) model to a smaller model with fewer degrees of freedom (DOFs) corresponding to the measured DOFs. Detailed formulation leading to the development of the posterior probability density function (PDF) is presented based on the current framework. Transitional Markov chain Monte Carlo (TMCMC) and Metropolis-within-Gibbs (MWG) sampling methodologies are used to approximate the posterior PDF. The effectiveness and efficiency of the proposed methodology are demonstrated using two numerical examples.
... Koruk and Sanliturk [4] confirmed the results obtained by Morzfeld et al., thereby highlighting the importance of replication and verification in scientific research. To overcome this limitation, complex eigenvectors of diagonally dominant systems can be generated using simple mathematical formulations considering the imaginarity of the eigenvectors due to the off-diagonal terms of modal damping matrix, as demonstrated in [28,30,40]. ...
This study explores alternative methods for decoupling non-classical linear damped systems in structural dynamics, aiming to replace the computationally expensive exact state-space method. Two methods are investigated: the lightly non-classical damping method (modal approximation method) and Adhikari's first-order algorithm. The lightly non-classical damping method is a cost-effective approach based on undamped eigenanalysis. However, it is less efficient for highly non-classical damping systems. In contrast, Adhikari's first-order algorithm offers a more efficient and low-cost solution, utilizing undamped eigenanalysis and employing Neumann series and Galerkin error minimization techniques. The suitability of these methods is determined by two diagonality dominance indices that identify the non-classicality of the system damping. For lightly non-classical damped systems, the study proposes a formula that uses undamped eigenvectors to enhance decoupling performance and generate complex eigenvectors. On the other hand, for highly non-classical damped systems, Adhikari's first-order algorithm is recommended. To evaluate the effectiveness of these methods, a four-degree-of-freedom frame system with three non-classical damping properties is examined. Additionally, the study suggests a new formulation of a subspace algorithm that combines both methods. The research findings contribute to the field of structural dynamics by providing an efficient and low-cost strategy for decoupling non-classical linear damped systems.
... However, experimental evidence to support the use of classical damping to characterize damping in a physical system is lacking [33]. Most real-life structures show that the criteria for classical viscous damping are not met, and these structures have complex modes rather than real modes [34,35]. Very few works in the literature have utilized the complex-valued modal data arising due to non-classical damping for FE model updating. ...
An approach for the Bayesian model updating of a linear dynamic system using complex modal data identified from dynamic test data is proposed in this paper. Very few works have utilized complex modal data for Bayesian model updating, and these works consider only the Most Probable Value (MPV) of the modal parameters as the modal data. On the other hand, the present work considers the posterior uncertainties of the modal parameters along with the MPVs as the modal data. Additionally, dynamic reduction has been applied to downsize the full system model to a reduced model with master Degrees of Freedom (DOFs) of the reduced model confined to the observed DOFs. Dynamic reduction facilitates updating of structural parameters without the requirement of mode matching. Additional uncertain parameters in the form of system modal frequencies, damping ratios, and partial mode shapes are introduced to establish a link between the structural model parameters and the modal data through the eigenvalue equation. Detailed formulation leading to the development of the posterior Probability Density Function (PDF) is presented, and a new Metropolis-within-Gibbs (MWG) sampler is proposed to simulate samples from the posterior PDF. Furthermore, a formulation is presented for evaluating the probability of damage based on the posterior samples obtained using the proposed approach from the structure's undamaged and possibly-damaged state. The proposed approach is validated and demonstrated comprehensively through simulated examples and an experimental study. Lastly, the performance of the proposed approach is compared with an alternative approach which is developed by integrating out the partial system mode shape from the posterior PDF.
... However, there isn't enough experimental evidence to back up the classical damping assumption in physical systems [18]. The majority of physical systems demonstrate that the classical damping criteria are not met, and as a result, these systems exhibit complex modes as opposed to real modes [19,20]. Complex mode shapes can be considered as propagating waves with no stationary nodes. ...
... Systems with much localized damping, such as automobile bodies with shock absorbers, soil-structure systems, and systems fitted with viscous dampers have complex modes. The existing output-only methods for complex modal parameter identification, such as those based on the Hilbert-Huang transform [21], wavelet transform [22], and blind source separation [23][24][25] have been shown to be effective for free vibration [19]. The use of blind source separation has been extended to analyze systems subjected to non-stationary excitations [26]. ...
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... However, the experimental modeshapes can also be used to identify the complete damping matrix provided updated mass and stiffness matrices are known. Lancaster's formula [47][48][49] for the direct damping matrix identification is given as follows: ...
Complex engineering systems are generally manufactured using joints. This paper proposes a methodology for accurately identifying the stiffness and damping properties of welded stiffened plates. The results of the substructure based finite element models are compared with the actual experimental data. Discrepancies have been observed between the finite element and experimental results. Eigensensitivity studies have been carried out to select the updating parameters. The stiffness and viscous damping properties of the stiffened plate are identified using model updating. The accuracy of the proposed methodology has been validated by matching the predicted finite element FRF with the experimental FRF.
... The MAC represents a suitable tool [41,42], to define the similarity of the two modal optimization results, and to reorder the eigenfrequencies numbering scheme if necessary. The general computation of MAC ∈ ℝ p×p for two eigenmodes a and b of a modal space Ф ∈ ℝ n×p is calculated with Eq. (9) where ( * ) denotes the Hermitian operator [42,43]. The displacement amplitudes are compared for each eigenmode between the current optimization iteration φ a,k and its previous φ b,k−1 . ...
Brake squeal reduces comfort for the vehicle occupants, damages the reputation of the respective manufacturer, and can lead to financial losses due to cost-intensive repair measures. Mode coupling is mainly held responsible for brake squeal today. Two adjacent eigenfrequencies converge and coalesce due to a changing bifurcation parameter. Several approaches have been developed to suppress brake squeal through structural changes. The main objective is to increase the distance of coupling eigenfrequencies. This work proposes a novel approach to structural modifications and sizing optimization aiming for a start at shifting a single component eigenfrequency. Locations suitable for structural changes are derived such that surrounding modes do not significantly change under the modifications. The positions of modifications are determined through a novel sensitivity calculation of the eigenmode to be shifted in frequency. In the present work, the structural changes are carried out on a beam and a brake caliper. Selected eigenfrequencies are shifted while the frequencies of the other eigenmodes are simultaneously fixed. Experimental investigations for the brake caliper validate the numerical findings and the applicability as well as efficiency of the proposed methods.
... 6 Bajric and Høgsberg (2018) identified the damping and complex modes in structural vibrations. 7 Xu et al., (2020) detected structural damage and modal parameter estimation with free response measured by continuously scanning LDV system. 8 ...
... Lamb waves). These vibration modes are specific to the used piezoelectric materials and are still at present the subject of research [12][13][14][15][16]. ...
The choice of piezoelectric material for a mechanical resonator sensor development is essential in terms of sensitivity, compatibility with the environment in which it will operate (high temperatures, environment acidity, …), complexity of the manufacturing processes implemented and costs involved. In this paper, piezoelectric detection is studied to understand mechanical resonator sensor operation principle and express it into a mathematical model. Validation by simulation tests of the developed model of measurement accuracy and measurement error as a function of relative vibration movement is performed. Applying this model, sensor characteristics and performance will be improved therefore, a new mechanical resonator sensor design can be proposed. The aim of these improvements is to obtain more accurate results and provide accurate information on vibratory level. A comparative study is conducted to show the effectiveness of the obtained results compared to the literature. These results have also demonstrated that a suitable and appropriate choice of damping rate improve the accelerometer operation and enhances the vibratory analysis technique.
