Modal-based Method Research Articles

A modal-based method for blast-induced damage assessment of reinforced concrete columns: Numerical and experimental validation

Reinforced concrete (RC) columns play a crucial role in the overall performance of modern RC frame buildings under blast loading. The RC columns under blast loads might experience various irreversible consequences such as concrete cracks, crushing and spalling, changing the status of both modal properties and load-carrying capacity. In this paper, the blast-induced damage assessment of the RC columns involving the measurement of modal parameters was introduced and validated. Firstly, a modal-based method for blast-induced damage assessment of RC columns was proposed, in which the damage is characterized as the reduction of axial bearing capacity and further expressed as the explicit function of the vibration frequencies and the mass-normalized displacement mode shapes. Then, the finite-element analysis program LS-DYNA was employed to establish the numerical model of RC columns under axial compression loads and lateral blast loads. The residual load-bearing capacities and modal properties of the RC columns were illustrated. By using the numerical results collected in the present study, the validity of the proposed damage assessment method was discussed and proved numerically. Further, field explosion tests and experimental modal tests were conducted, and the results showed that the modal-based damage assessment method exhibited remarkable agreement with the blast damage determined by residual bearing capacities. Based on the numerical and experimental results, the proposed modal-based damage assessment method is applicable for the non-destructive evaluation of blast-induced damage of RC columns.

Hybrid modal-machine learning damage identification approach for beam-like structures

Data-driven damage detection methods are widely researched and implemented due to the availability of advanced sensing and cloud technologies, where machine learning models are used to process the various data collected from the sensors for damage diagnosis. Supervised methods have shown to be accurate in identifying damages given that they are trained with sufficient historical damage samples. As damage events are rare, supervised methods often face the cold-start problem, that is, insufficient labelled damage samples to initiate model training. Unsupervised methods do not require training and have shown to be practical up to detecting damage presence, but prior knowledge or labelled historical damage samples are still required as references to further locate and quantify the damage. To solve the cold-start issue, a modal-based method, which relates the modal parameters with damages, is proposed to bridge the gap between the unsupervised and supervised methods. Therefore, this work proposes an integration strategy of a modal-based method into a hybrid machine learning-based method for damage detection and localisation of beam-like structures. A two-stage scheme with unsupervised, supervised, and modal-based methods is proposed. The first stage detects damage presence using PCA-FRF as the damage-sensitive feature in an unsupervised manner, and the second stage locates the damages using the first mode shape. Mode shape assessment-based damage localisation is performed at the second stage when no trained model is available due to zero or limited damage samples for training to solve the cold-start and manual labelling issue. Supervised model training starts once there are at least two classes of damage, and the trained model is only used for damage localisation when it surpasses the 90% validation accuracy benchmark. Results showed accuracies of 100%, 100%, and 95.85% for the first stage unsupervised detection, second stage mode shape assessment-based damage localisation and the supervised model-based damage localisation, respectively.

Modal-Based Analysis for Aiding 3D Elastic Metastructure Design

The engineered topological structures of the unit cell endow elastic metamaterials (EMMs) with the extraordinary capability to attenuate elastic waves. In real-life scenarios, a practical EMM (i.e., metastructure), consisting of a limited number of unit cells, is the truncation of the infinite EMM, which may detriment seriously the attenuation capability. To understand the mechanism behind the detriment, an efficient way to evaluate and analyze the stopbands for metastructures is essential. Instead of relying on the commonly adopted frequency response analysis to characterize stopbands for metastructures, which are sensitive to frequency sweeping steps and short of information of truncated resonance affecting wave attenuation performance, a novel modal-based method (MM) is proposed to assess the wave attenuation of 3D metastructures. Specifically, the modal-based analysis scheme incorporating the modal superposition principle and a freshly developed layer-based strain energy ratio (SER) measurement method is proposed. Through employing the MM, the opening and ceasing of the stopbands are evaluated by characterizing dominant eigenmodes developed in metastructures. Accordingly, the influences of eigenmodes induced by different mechanisms on wave attenuation performance are investigated, and the stopband formation mechanisms are elaborated. Furthermore, under the guidance of a freshly proposed modal-based design framework, metastructures with modified geometrical parameter settings (GPSs) are demonstrated with enlarged normalized bandwidths (NBs) at lower frequencies, which show improved applicability in multiple engineering disciplines.

Real-Time Virtual Sensing for Dynamic Vibration of Flexible Structure via Fiber Bragg Grating Sensors

Real-time monitoring of the dynamic vibrations of beam-type structures indicates important implications in aerospace engineering applications, in which the additional sensing information about vibration parameters, including displacements, rotational angles, angular rates, as well as accelerations, facilitates the improvement of the system’s safety. This article proposed a virtual sensing method with the usage of fiber Bragg grating (FBG) strain sensors to directly estimate the vibration parameters from the measured strains. The algebraic relationships of displacement–strain and rotation–strain on the sensor’s installing points as well as the boundary points are obtained based on the Birkhoff interpolation method, and meanwhile, the explicit expressions on angular rates and accelerations are derived. Furthermore, a tracking differentiator is used to recover the strain signals and their first- and second-order derivatives from the noisy measurements in order to real-time estimate the angular rates and accelerations of the vibrating beam. Differing from the classical modal-based method, complex modal decomposition is not required in the proposed method. The comparison of numerical simulation results reveals that the proposed method has advantages over the modal-based method in accuracy and efficiency. Furthermore, the laboratory experiments indicate that the proposed method is effective for real-time estimating vibration parameters from the noisy measured strains.

Prediction and analysis of sound field in long enclosures with a modal-based method

In this paper, the efficacy of porous ceiling treatment to reduce noise levels inside a typical tunnel is examined with a validated modal-based prediction method. It is found that, for a point source, the effect of increasing porous ceiling thickness on sound pressure level (SPL) attenuation along the tunnel is limited. A porous ceiling with thickness of 0.3 m is comparable with an infinite porous ceiling in middle and high frequency ranges. For a line source, the effect of ceiling thickness on SPL reduc- tion in this typical tunnel is limited. Sound pressure level reduction of 4 dBA is real- ized with 0.3 m porous ceiling, which is the same as infinite ceiling and only 1 dBA smaller than the theoretically optimized value. These results suggest that, in the event only ceiling treatment is considered, 0.3 m porous material is sufficient for noise re- duction in this typical tunnel.

Efficient modal-based method for analyzing nonlinear aerostatic stability of long-span bridges

An efficient method for analyzing the nonlinear aerostatic stability of long-span bridges using a modal formulation is proposed. First, a simplified linear version defined as modal-based approach is introduced, which evaluates the bridge displacements using the modal properties of the bridge and speeds up the assessment of the critical velocity by applying a root-finding algorithm. Then, this scheme is further developed to identify the instability limit point in nonlinear structures by combining nonlinear FEM analyses with the linear version of the modal-based approach and a root-finding algorithm in an outer loop. The effects of considering the three components of the wind loads, the stays sag effect, structural nonlinearities, aerodynamic nonlinearities, and the initial wind angle of attack, are analyzed and discussed. The computational advantages of the proposed method and its accuracy are demonstrated through three application examples, including a simplified linear 1 DoF system, and linear and nonlinear FEM models of a full cable-stayed bridge. The collapse mechanism of the cable-stayed bridge is driven by the stiffness degradation of the stays caused by the deck vertical upward displacements. It has been found that the influence of the nonlinear aerodynamic features is very low when the structural nonlinearities control the bridge collapse.

Bidirectional Beam Propagation Method Based on Axi-Symmetric Full-Vectorial Finite Element Method

We first present a bidirectional beam propagation method (BiBPM) based on axially symmetric full-vectorial finite element method (Axi-FVFEM). The Axi-FVFEM based BiBPM is a more versatile method than previous scalar BiBPM for axi-symmetric structure. It has potential to offer more efficient analysis than widely used FEM or finite difference based techniques, especially for a large scale waveguide such as a long periodic structure. An air gap in an axi-symmetric waveguide and a fibre Bragg grating (FBG) are analyzed using the Axi-FVFEM based BiBPM for numerical validation, and we compare the numerical results with those of a 3D finite-difference time-domain (FDTD) method and a modal-based method.

Modal Rotations: A Modal-Based Method for Large Structural Deformations of Slender Bodies

This study presents the derivation and application of the modal rotations method (MRM), a novel framework for the computation of large deformations of complex wing-like structures using a modal approach. The method targets static analyses of slender structures, accounting for large-deformation nonlinearity. The input for the analysis is linear modal finite-element-based data. However, the MRM uses modal rotation data, rather than deformation modes. The modal rotations are used for the solution of the nonlinear kinematic problem along with an iterative load-correction procedure that accounts for the geometric stiffening. The method is suitable for cases where the generation of an equivalent beam model is impossible or requires considerable effort, such as structures of multiple different sections, structures having abrupt geometrical changes, and lattice structures. The MRM is verified with three test cases. The first is a simple geometry, a one-dimensional symmetric beam loaded in one plane. The second and third test cases are three-dimensional built-up structures aimed to challenge the method with abrupt geometric changes, orthotropic material coupling, and nontrivial geometries. All test cases yield accurate results compared with nonlinear finite element analyses. Parametric studies present the dependency of the results on different computational parameters and the load magnitude.

A simplified modal-based method to quantify delamination in carbon fibre-reinforced plastic beam

ABSTRACTThis study contributes to our knowledge about the detection and recognition of carbon fibre-reinforced plastic (CFRP) composites by using a modal analysis technique. Delamination was artificially induced during the manufacturing process of CFRP beams so that the dimension and location of the artificial delamination were known. The first three longitudinal bending modes were considered for detecting, locating, and evaluating the delamination. Contrary to what is commonly practised, the sensor placed on the CFRP structure had a large mass in order to contribute actively to restricting the delamination and hence modifying the structure’s natural frequencies, thus making it much easier to monitor its health condition. The technique was implemented on a beam with a delamination gap of 9% of the beam’s length, positioned within the span of the beam. The results demonstrate that the proposed technique can be effective not only to locate the delamination but also to measure its extent accurately.

Radiation Energy and Mutual Coupling Evaluation for Multimode MIMO Antenna Based on the Theory of Characteristic Mode

This paper presents a novel method to evaluate radiation energy and mutual coupling in multimode antennas. Based on the theory of characteristics mode, how much each mode occupies the radiation and the mutual coupling from each feeding port is calculated with the modal energy occupied coefficients. Furthermore, the linear transformation of feeding network in multimode antenna system has been adopted to complete the modal-based method. Then, this method is utilized to analyze and decrease the mutual coupling between feeding ports. Hence, a hexagonal wideband antenna is proposed with its evolution process and measured to validate the proposed method. The presented hexagonal antenna is a four-port multimode antenna consisting a planar hexagonal plate, vertical tapered baluns, and feeding network at the ground plane. The whole antenna works in 3–6 GHz and all its four ports are well matched with high port-to-port isolation.

Active control of bending waves propagating in an orthotropic rectangular panel

Due to the recent advances in microprocessors, active control of a flexible structure has become a realizable method for suppressing the vibration. Conventional modal-based method encounters difficulties in controlling a distributed parameter structure, since such a structure has an infinite number of vibration modes. To overcome this problem, active wave control has been studied in recent years. This paper presents an active wave control of an orthotropic rectangular panel by expanding the conventional active wave control method for a two-dimensional structural case. First, a transfer matrix method for a rectangular panel is introduced to describe the wave dynamics of the structure. This is followed by the derivation of feedforward control laws for absorbing reflected waves or eliminating transmitted waves. In the proposed method, the control laws are including the modal actuation scheme for uncontrolled direction. Then, from a viewpoint of numerical analyses, basic properties of the proposed method are verified. It is found that the reflected wave absorbing control enables the inactivation of all vibration modes in the controlled direction and the transmitted wave eliminating control enables the generation of an almost vibration-free state.

A modal-based multiscale method for large eddy simulation

We introduce a modal spectral element method which employs the variational multiscale approach for large eddy simulation. The method is sufficiently general for application to complex flow geometries, but requires relatively few degrees of freedom for a given mesh and polynomial order. We investigate the performance of the method for fully-developed turbulent channel flow. It is shown that large increases in accuracy can be obtained when the basic dynamics of near-wall coherent structures can be represented using the portion of the modal basis which is free of the subgrid-scale model. This implies that in spite of its lack of orthogonality, the modal basis provides sufficient scale separation to exploit the advantages of the variational multiscale approach.

Hydroelastic behaviour of a conical shell impacting on a quiescent-free surface of an incompressible liquid

Hydroelastic effects are simulated by coupling a hydrodynamic Wagner model to a linear model of elasticity for thin shells. Applications are done for a cone falling on a flat-free surface of an incompressible liquid. Both hydrodynamic and structural models are linearized on the basis of a flat disk approximation. This is justified when the deadrise angle is small. In the hydrodynamic Wagner model, the main task is to evaluate the time-varying expansion of the wetted surface. The coupling with the linear model of elasticity is achieved via a modal-based method. This means that the hydrodynamic variables must project onto the family of eigenfunctions; this is the second main difficulty of the present problem. The coupled problem is solved mainly analytically. Special attention is paid to the energy conservation law. In particular, it is shown that kinetic energy evacuated in the jet plays as significant a role in the distribution of energy as the kinetic energy transmitted to the fluid or the kinetic and potential energies of the elastic shell. The importance of elasticity is discussed by comparing the rigid and elastic behaviours for free drop tests. A parametric study shows the influence of each parameter: thickness, deadrise angle and drop height. Comparisons with available experimental pressure data show a reasonable agreement. On the basis of this work, lines of future research are outlined.

ACOUSTIC RADIATION FROM A FINITE-LENGTH SHELL WITH NON-AXISYMMETRIC SUBSTRUCTURES USING A SURFACE VARIATIONAL PRINCIPLE

A modal-based method is developed to analyze the acoustic radiation of axisymmetric submerged shells of finite length with non-axisymmetric internal substructures, subjected to time-harmonic loads. In this method, a variational principle is used to derive impedance relations between the surface pressure and the surface velocity of the shell. These impedance relations are combined with the structural equations of motions based on a Lagrange energy formulation to form a complete set of equations for the fluid–structure interaction problem. Fourier series expansions are used to represent the circumferential dependence of the surface pressure and velocity. The method is demonstrated for two different configurations of substructures: circular ribs supporting length-wise beams and a spatially and modally dense array of oscillators. Since the substructures couple the circumferential modes, a large system of equations must be inverted. A matrix decomposition technique is used to reduce the size of the system of equations of the first example. For the second example, the asymptotic limit for an infinite number of substructures is developed using an integral form for the substructure impedance. It is shown that the Monte Carlo simulations for the oscillator substructures converge to the asymptotic results. It is also shown that, below the ring frequency, oscillators can induce damping that is more effective in reducing far-field radiation than high loss factors in the shell.

Acoustic radiation from a finite-length shell with substructures subjected to an impulsive load

A method is presented to obtain the impulsive response and the acoustic radiation of submerged shells of finite length. Solutions in the frequency domain are first evaluated by a modal based method and converted to the time domain using a fast Fourier transform technique. Numerical examples are presented for a cylindrical shell of finite length with endcaps, with and without bulkheads. The method is validated by comparing surface pressure response of the empty shell with that of an infinite, fluidloaded shell. It is demonstrated that the near-field pressure disturbances arise from elastic waves propagating on the shell and from the direct radiation from the driving force. The amount of energy input to the system by the driving force and the energy radiated to the far-field are calculated. Results show that most of the energy is dissipated in the shell by damping. It is also shown that shells containing bulkheads can radiate more energy than empty shells because the bulkheads generate supersonic, radiating flexural waves.

Acoustic scattering from a hemicapped cylindrical shell with substructures

This paper extends the study on acoustic radiation of axisymmetric submerged shells of finite length to the scattering problem. A modal-based method is developed to determine the scattering from finite shells insounified by plane incident waves. A surface variational principle provides the acoustic impedance relationship between the surface pressure and displacement fields, which are expanded into a series of shape functions. Lagrange multipliers are introduced in a Lagrange formulation of the dynamic analysis to account for the interaction between the shell and internal substructures. Surface and far-field pressures scattered from the elastic shell are calculated. It is found that, at certain angles, the scattered pressure field has pronounced peaks, particularly as the incident frequency increases. A wave number domain analysis is performed on the surface displacement field. A dominant wave number is found which accounts for the high directivity pattern of the far field. The effect of substructures on the far field is also investigated.

Nonaxisymmetric vibration and acoustic radiation of a submerged cylindrical shell of finite length containing internal substructures

A modal-based method is developed to analyze the acoustic radiation of axisymmetric submerged shells of finite length with internal substructures, subjected to nonaxisymmetric time-harmonic loads. In this method, a variational principle is used to determine the relationship between the surface pressure and displacement of the shell, and Lagrange multipliers are used to account for the connections between the shell and the substructures. Fourier series expansions are used to represent the circumferential dependence of the surface pressure and displacement. The method is demonstrated by an extended analysis of a cylindrical shell with hemispherical elastic endcaps containing circular bulkheads. The dominant flexural wave numbers are identified from helical wave spectra. It is shown that the wave numbers of the dominant flexural waves are the same as those found in an infinite, fluid-loaded cylindrical shell. The locus of wave numbers of the dominant flexural waves lies outside of the sonic cone for low to mid frequencies. However, it is shown that bulkheads can create amplification of the flexural waves with wave numbers within the sonic cone. This is confirmed by a computation of the net power radiated by the shell: The shell with bulkheads radiates more energy than the empty shell for ka≳0.5, where k is the acoustic wave number and a is the radius of the cylindrical shell.

A new approach to modal-based structural dynamic model updating and joint identification

In this paper, a new modal-based method for the two tasks of structural dynamic model updating and identification of joint models in structural assemblies is introduced. The method is formulated in detail and its theoretical and computational capabilities and shortcomings are discussed. The performance of the new method is assessed through a series of case studies and the effects of various parameters on the results are discussed. Also, a sensitivity study is undertaken to examine the method’s performance in the presence of measurement noise.

The effect of substructures on the acoustic radiation from axisymmetric shells of finite length

A modal-based method is developed to analyze the acoustic radiation of shells of finite length with internal substructures subjected to time-harmonic loads. In this method, a variational principle is used to determine the relationship between the distribution of velocity and pressure over the surface of the shell, and Lagrange multipliers are used to account for the interaction forces from the substructure motion. To identify the dominant physical characteristics of the fluid/shell/substructure response, a singularity decomposition is introduced. This decomposition uses the singular points of the response in complex wave-number space that are closest to the real axis. The response contributions from these singular points are found to be highly fluctuating, yet analytically simple functions of frequency. The remaining or residual response is a slowly varying function of frequency. To reduce computational effort, the singularity decomposition is combined with a frequency window method. The method is demonstrated for a cylindrical shell with elastic, hemispherical endcaps and internal circular panels. The effects of the substructure are illustrated by comparing the response of the shell/substructure system with corresponding results for an empty shell. It is shown that the substructures can significantly alter the near- and far-field acoustic response.

The effect of substructures on the acoustic radiation from axisymmetric shells of finite length

A modal-based method is developed to analyze the acoustic radiation of shells of finite length with internal substructures subjected to time-harmonic loads. In this method, variational analysis is used to solve for the shell surface velocity and pressure, and Lagrange multipliers are used to determine the substructure motion. To gain insight into the physical characteristics of the fluid/shell/substructure system, a singularity decomposition is introduced. This decomposition is based on the response singularities of the system in complex wave-number space that are closest to the real axis. It is then shown how the decomposition leads to a frequency window method to reduce computational effort. The method is demonstrated for a cylindrical shell with hemispherical endcaps and an internal circular panel. The effects of the substructure are illustrated by comparing the response of the shell/substructure system with corresponding quantities for an empty shell. [Work supported by ONR.]