Modal Expansion Method Research Articles

Dynamic force identification considering modeling errors using modal expansion method and relevant vector regression algorithm

Accurate identification and estimation of external forces on launch vehicles, aircraft, and other in-service engineering structures plays a vital role in structural design and health monitoring. Considering structural modeling errors, this paper develops a novel force identification method based on modal expansion and relevant vector regression (RVR). Initially, the incomplete and noisy experimental modal data are used to modify the stiffness and mass matrices of the finite element (FE) model, reducing the impact of modeling errors on force identification accuracy. Subsequently, to account for noise disturbances and potential information about unknown initial conditions in the measured acceleration responses, the external forces and the unknown initial conditions are respectively expanded using trigonometric functions and the mode shapes based on the function fitting technique. On this basis, the force identification equation is established for the updated model. The RVR algorithm is integrated into modal expansion and force identification. Firstly, the mode shapes of the FE model are utilized to expand the incomplete and noisy experimental mode shapes to their full degree of freedom, significantly enhancing the accuracy of the model updating. Secondly, by combining the measured acceleration responses, the base coefficients can be solved sparsely, effectively identifying the dynamic forces even when unknown initial conditions are present.

Damage detection method for square steel tube based on CS-NME algorithm via ultrasonic guided waves

The utilization of ultrasonic guided wave (UGW) technology has garnered considerable attention in non-destructive testing field, particularly for steel components. However, for square steel tubes with noncircular cross sections, this method encounters severer frequency dispersion, which hampers the accuracy of damage identification. To address this issue, the approach utilizing the normal mode expansion (NME) method combined with cuckoo search (CS) algorithm to develop a layout strategy for ultrasonic transducer position and length is proposed, which enables single mode excitation of UGW while reducing the interference from redundant clutter signals, thereby enhancing the accuracy of damage detection. The efficacy of the proposed approach is evaluated via numerical simulations and laboratory experiments. The results show that this method can significantly reduce the interference of other modes by optimizing the location and length of ultrasonic transducers, and successfully detect the hole and crack damage on the square steel tube.

Computation Theory of Large-Scale Partially Coherent Imaging by the Modified Modal Expansion Method

The numerical calculation of partially coherent imaging involves a fourfold integral, which is numerically complex and impracticable to be calculated directly. The use of coherent-mode decomposition (CMD) can make this problem more manageable but finding the coherent-modes for complicated partial coherent fields (without already-known coherent-mode expansion) are rather computationally intensive. In this letter, a modified modal expansion method is proposed, which significantly reduces the requirement of computational resources. The propagation of partial coherence in imaging systems with extremely large sampling number could be handled by an ordinary computer. A comparison between the new method and the traditional method in terms of memory resource requirements and computational time consumption is also detailed in this article. We will also show that this method could deal with the anisoplanatic imaging cases while maintaining the same computational efficiency.

Insight into local defect resonance and its in-situ measurement based on flexible nano-composite sensors

The local defect resonance has been proven to be effective for damage detection, and the methods based on LDR for damage evaluation have demonstrated appealing capabilities of selective assessment and applicability to complex structures. To explain the generation of LDR, models based on vibration theories have been developed in which the defect boundaries are assumed to be fixed. Nevertheless, existing models can only give an accurate prediction of frequency for fundamental LDR mode owing to the deviation of their assumptions from the reality. In addition, the measurement of LDR is currently limited to noncontact technologies, hampering its application to in-situ and online monitoring of inspected structures. To overcome these deficiencies, an analytical model is proposed in this investigation to gain an insight into the generation of LDR from the perspective of wave reflection at defect boundaries and wave superposition. In this model, the interaction of guided waves with defect is scrutinized using a normal mode expansion method to obtain the phase shift of the reflected waves at the defect boundaries. On this basis, the formation of standing waves is analyzed to interpret the occurrence of the resonance of guided waves, whereby the relation of the LDR frequencies and the defect parameters can be obtained. On top of this analytical investigation, the sensing of the LDR using a novel nano-composite sensor is attempted. Flexible and lightweight, these sensors can form a dense sensing network, with which the LDR response in the inspected region can be perceived. Using the LDR-based method and nano-composite sensors, this approach can give a damage characterization in an in-situ and online manner for complex structures. Experimental validations of the proposed approach is performed in which delaminations in composite structures are identified and assessed using the LDR response perceived with the nano-composite sensors.

Vibration control of thin shallow paraboloidal shells with light-activated shape memory polymers

With the applications of thin-walled paraboloidal shell structure in aviation, aerospace and communication systems, smart-structure based active vibration control becomes essential to improve the performance of thin-walled parabolic structures. Light-activated shape memory polymer (LaSMP) is a novel actuator material that exhibits dynamic Young's modulus and strain properties under ultraviolet (UV) light exposures, which can be used for non-contact vibration control. This study focuses on investigating the vibration control performance of a shallow parabolic thin shell, with simply supported boundary conditions, with laminated LaSMP patches. Based on the LaSMP strain model, the vibration control effect of LaSMP patch on the thin shell is discussed. A finite element model of the simply supported parabolic thin shell is developed to obtain the natural frequencies. The modal expansion method is employed to investigate the vibration control of the shallow parabolic thin shell laminated with LaSMPs, and independent modal responses are calculated. According to the different control mode, the effects of LaSMP actuators on vibration control of shallow paraboloid shell are studied in three cases with varying actuation position and coverage area of LaSMP actuators. The results indicate that LaSMP can control the vibration of the thin shell by reducing the vibration amplitude and the final vibration control effect is determined by the mode shape, the initial LaSMP strain, and the location and area covered by LaSMP actuators. By establishing the model between the LaSMP actuator force and the attenuations in modal amplitude, this study provides an analytical tool for the application of LaSMPs in vibration control of flexible structures.

Unified model modal expansion method for full field reconstruction and inversion under mechanical and thermal loading

In this paper, a novel method termed UMM-T&S (Unified Model Modal reconstruction and inversion method for Temperature field and Structural field under limited sensors) is proposed. This method aims to enhance high-precision structural performance inversion and reconstruction under mechanical and thermal loading conditions, particularly in the context of structural health monitoring within the aviation and aerospace sectors. The approach begins by deriving the “temperature modal” solution of the heat conduction equation, drawing upon the modal expansion techniques used in multi-degree-of-freedom systems. This is followed by constructing a parameter mapping relationship within the unified model, enabling the solution of both the structural field (encompassing strain and displacement) and the temperature field through a single structural model modal expansion. Furthermore, the paper details a numerical validation using two different temperature fields and a thermal loading experimental verification on a stiffened plate under five distinct working conditions. The proposed UMM-T&S method yields lower errors (ranging between 10%-40%) and more stable performance across various complex working conditions, in comparison to existing methods. The analysis also delves into the sources of errors and their cumulative impacts.In conclusion, the innovative UMM-T&S method offers a swift, precise, and robust solution for the intricate task of reconstructing and inverting structural performance under combined mechanical and thermal stresses. The method's efficacy and potential are substantiated through experimental validation and comparative analysis against an established methodology, underscoring its capability to effectively tackle the challenges faced in structural monitoring under such demanding conditions.

Fatigue Analysis of a Jacket-Supported Offshore Wind Turbine at Block Island Wind Farm.

Offshore wind-turbine (OWT) support structures are subjected to cyclic dynamic loads with variations in loadings from wind and waves as well as the rotation of blades throughout their lifetime. The magnitude and extent of the cyclic loading can create a fatigue limit state controlling the design of support structures. In this paper, the remaining fatigue life of the support structure for a GE Haliade 6 MW fixed-bottom jacket offshore wind turbine within the Block Island Wind Farm (BIWF) is assessed. The fatigue damage to the tower and the jacket support structure using stress time histories at instrumented and non-instrumented locations are processed. Two validated finite-element models are utilized for assessing the stress cycles. The modal expansion method and a simplified approach using static calculations of the responses are employed to estimate the stress at the non-instrumented locations-known as virtual sensors. It is found that the hotspots at the base of the tower have longer service lives than the jacket. The fatigue damage to the jacket leg joints is less than 20% and 40% of its fatigue capacity during the 25-year design lifetime of the BIWF OWT, using the modal expansion method and the simplified static approach, respectively.

Ultra-high Q resonances based on zero group-velocity modes accompanied by bound states in the continuum in 2D photonic crystal slabs

Optical resonators made of 2D photonic crystal (PhC) slabs provide efficient ways to manipulate light at the nanoscale through small group-velocity modes with low radiation losses. The resonant modes in periodic photonic lattices are predominantly limited by nonleaky guided modes at the boundary of the Brillouin zone below the light cone. Here, we propose a mechanism for ultra-high Q resonators based on the bound states in the continuum (BICs) above the light cone that have zero-group velocity (ZGV) at an arbitrary Bloch wavevector. By means of the mode expansion method, the construction and evolution of avoided crossings and Friedrich-Wintgen BICs are theoretically investigated at the same time. By tuning geometric parameters of the PhC slab, the coalescence of eigenfrequencies for a pair of BIC and ZGV modes is achieved, indicating that the waveguide modes are confined longitudinally by small group-velocity propagation and transversely by BICs. Using this mechanism, we engineer ultra-high Q nanoscale resonators that can significantly suppress the radiative losses, despite the operating frequencies above the light cone and the momenta at the generic k point. Our work suggests that the designed devices possess potential applications in low-threshold lasers and enhanced nonlinear effects.

Induced effects of a rigid boundary on a pulsating spherical source

This work develops a comprehensive analytical formalism to study the boundary-induced effects on a pulsating spherical radiating source near a rigid planar boundary in an inviscid fluid. Using the modal multipole expansion method, the image method and the translational addition theorem for spherical wave functions, exact closed-form expressions are derived for the dimensionless axial acoustic radiation force function as well as the radiation, amplification and extinction energy efficiency factors. The source-boundary distance and the dimensionless size parameter are especially addressed in the numerical simulations for spherical sources with different vibration modes. It is shown that the axial radiation force is positive or negative, depending on the choice of parameters. Moreover, cases when the extinction acoustic power completely vanishes are predicted theoretically for dipole and quadrupole vibrations, which can be used to achieve acoustic invisibility in acoustic cloaking devices and acoustic stealth technologies. Combination of more than one vibration modes is also investigated to better simulate the acoustic manipulation in practical applications. This work can help us understand the underlying mechanism in manipulating the active spherical radiating sources in a bounded field.

Method Comparison for Simulating Non-Gaussian Beams and Diffraction for Precision Interferometry.

In the context of simulating precision laser interferometers, we use several examples to compare two wavefront decomposition methods-the Mode Expansion Method (MEM) and the Gaussian Beam Decomposition (GBD) method-for their precision and applicability. To assess the performance of these methods, we define different types of errors and study their properties. We specify how the two methods can be fairly compared and based on that, compare the quality of the MEM and GBD through several examples. Here, we test cases for which analytic results are available, i.e., non-clipped circular and general astigmatic Gaussian beams, as well as clipped circular Gaussian beams, in the near, far, and extremely far fields of millions of kilometers occurring in space-gravitational wave detectors. Additionally, we compare the methods for aberrated wavefronts and their interaction with optical components by testing reflections from differently curved mirrors. We find that both methods can generally be used for decomposing non-Gaussian beams. However, which method is more accurate depends on the optical system and simulation settings. In the given examples, the MEM more accurately describes non-clipped Gaussian beams, whereas for clipped Gaussian beams and the interaction with surfaces, the GBD is more precise.

Guided wave propagation in a double-layer plate with a nonlinear spring-interface

This article derives the solution to the guided wave fields in a double-layer plate consisting of two sublayers. It is assumed that the two sublayers are linearly elastic. They are bonded together at their interface by a nonlinear adhesive layer of infinitesimal thickness. This allows us to propose a nonlinear spring-interface model. Based on such an idealized model for the double-layer plate, guided wave fields in the plate are solved using the modified normal mode expansion method. It is found that the nonlinearity of the spring-interface can generate resonant guided waves in the double-layer plate. Specifically, when certain conditions are met, mixing of two primary guided waves will generate resonant guided waves whose frequencies are either the sum or difference of those of the two primary waves. Amplitudes of such resonant mixed waves are proportional to the compliance of the nonlinear spring-interface. As a special case, if the two primary waves have the same frequency, a resonant second harmonic guided wave may be generated. In addition, the conditions that generate resonant mixed waves are identified. We believe that the results of this work provide the theoretical foundation on which nondestructive evaluation techniques using nonlinear guided waves can be developed to nondestructively evaluate, for example, the bond strength of thin coating on a substrate.

Low-frequency still-air acoustic inertia of inclined circular aperture in an infinite flat plate of finite thickness

The acoustic inertia of the canonical configuration of inclined circular aperture(s) in a finite-thickness plate at the low frequency limit is investigated under the inviscid still-air condition. A hybrid approach combining modal expansion and boundary element method is applied to calculate the effective length of the aperture as a quantitative characterisation of its acoustic inertia. These calculations, covering different inclination angles up to 75°, are performed for a single isolated aperture with a selected range of plate thickness and the periodic aperture arrays of aligned and staggered arrangements with a selected range of aperture spacing respectively. The results are validated by the simulations carried out with the commercial software COMSOL. The parametric studies of geometric dimensions included in this work provide representative results for typical acoustic related engineering applications of finite-length inclined apertures. A link for access to the MATLAB programmes implementing the calculations is provided for interested readers. As an example of application, the hybrid method described in this work is used for the acoustic modelling of a representative multi-perforated liner studied in the literature for the problem of thermo-acoustic instability. The results compared well against those obtained from the Computational Fluid Dynamics simulations reported in the literature.

Response prediction of cantilever plates via mode superposition method and combination method of beam functions

The present work is devoted to predicting the response of cantilever plates by expanding the vibration data at limited positions to the unmeasured regions with poor accessibility. The predictive algorithm is based on the mode superposition method (MSM) and the combination method of beam functions (CMOBF). Firstly, the vibration mode function (VMF) of the cantilever plate is deduced offline according to the CMOBF. According to the VMFs calculated offline and the responses measured at limited points, the mode coefficients are updated by the MSM in real time. Finally, the displacements or accelerations of target positions are predicted by the product of the mode matrix and the vector of mode coefficients. The predictive process is achieved in the absence of specific expression and distribution of external loads. Besides, different from traditional modal decomposition and expansion methods, the present work is not dependent on the finite element (FE) model or repeated iterations. The predictive process is convenient to carry out with a small computation time online. As some application examples, the predictive method is applied to the cantilever plates under concentrated and moving loads by simulations and experiments. Both the acceleration and displacement signals are taken into account and examined.

Strengths and weaknesses of the modal expansion method for perturbations calculations in nuclear reactor physics

Reactor design and uncertainties propagation request repeated execution of neutron diffusion calculations. To address this, one can rely on the Generalized Perturbation Theory which is restricted to single integral values such as reaction rate ratios. To overpass such limitations, the perturbed flux could be calculated with the modal expansion method. The difficulty to calculate more than one eigenvector has limited its use. Hence, modal expansion lacks a clear explanation of its strengths and weaknesses. Using recent advances in eigenvalues solvers, we have successfully applied modal expansion to the 2D-BIBLIS and IAEA-3D benchmarks. Our results are promising for fission and capture cross sections perturbations. Still, modal expansion is faulty for local perturbations or when it concerns the reflector. In terms of computational cost, modal expansion is more efficient than solving the direct problem with a calculation time divided by 8 for a basis of 20 eigenvectors.

Excitation response analysis of hollow cylinders under helical surface tractions for dominant flexural guided wave generation

Dominant flexural mode excitation of helical waves in hollow cylinders by applying helical surface tractions, equivalent to exciting obliquely propagating plate waves in the unwrapped plate, is reported. However, this approach cannot be generalized with theoretical underpinnings. The response of hollow cylinders under helical surface tractions is investigated using normal mode expansion method. Purity of the target flexural mode depends on the mode wave structures at excitation frequencies and the helix angle of the helical excitation. With helical surface tractions, only flexural modes with large circumferential and longitudinal displacements, usually torsional flexural modes, can be effectively generated. The target flexural mode and excitation frequency should be selected carefully for better purity. The helix angle for the target mode should diverge considerably from other modes, especially those with large circumferential and longitudinal displacements. The apparently oblique propagation of the flexural mode can also be explained theoretically. The helically excited flexural mode comprises two orthogonal wave structures with identical flexural modes that are out of phase in the propagation direction, causing periodically changing circumferential orientation of the combined wave structure and oblique wave front. Finite element simulations and experiments agree well with theoretical predictions.

A modal expansion method for simulating reverberant sound fields generated by a directional source in a rectangular enclosure

The prediction of reverberant sound fields generated by a directional source is of great interest because practical sound sources are not omnidirectional, especially at high frequencies. For an arbitrary directional source described by cylindrical and spherical harmonics, this paper developed a modal expansion method for calculating the reverberant sound field generated by such a source in both two-dimensional and three-dimensional rectangular enclosures with finite impedance walls. The key is to express the modal source density using the cylindrical or spherical harmonic expansion coefficients of the directional source. A method based on the fast Fourier transform is proposed to enable the fast computation of the summation of enclosure modes when walls are lightly damped or rigid. This makes it possible to obtain accurate reverberant sound fields even in a large room and/or at high frequencies with a relatively low computational load. Numerical results with several typical directional sources are presented. The efficiency and the accuracy of the proposed method are validated by the comparison to the results obtained using the finite element method.

Rigorous Modal Analysis of Chiral Nanoparticles

AbstractThe physical origins of the chiroptical response from nanostructures are highly demanded for chiral photonics. The chiroptical response of a nanoparticle, whether measured along specific incident directions of exciting waves (when immobilized onto a substrate) or in solution through averaging responses across all structure orientations, is believed to stem from the interplay between three key factors: geometrical chirality, far‐field chirality, and near‐field chirality. This study aims to understand the relationship among these factors using a rigorous quasinormal mode (QNM) expansion method. By decomposing circular dichroism (CD) spectra into individual QNM contributions, it is observed that the chiroptical response with respect to incident directions depends on the extent of field overlapping between the QNM and light. In contrast, the near‐field optical chirality (OC) is determined by the QNMs of the nanoparticle, independent of external excitations. As a proof‐of‐concept, a nanohelix metasurface is proposed that employs lossless gold and exploits the physics of bound states in the continuum to maximize the CD signal.

Implicit parametric modal expansion method for single-layer reticulated shells based on generative adversarial network

Single-layer reticulated shells (SLRSs) are a prevalent form of roofing used in public buildings, including stadium and exhibition centers. Structural health monitoring (SHM) systems have been installed on SLRSs to obtain modal parameters for their design, assessment, and maintenance. However, modifying a complex finite element (FE) model with modal information acquired from a limited number of sensors can be challenging, especially for SLRSs. It is necessary to expand the measurement mode to obtain the complete mode before modifying the FE model. This paper proposes an implicit parametric modal expansion method for SLRSs based on generative adversarial network (GAN) to accomplish modal expansion without prior knowledge of the stiffness and mass of SLRSs. A GAN is trained to establish mapping relationship under different combinations of stiffness and mass of the SLRS. Then the expanded modals can be obtained through the trained GAN after obtaining measuring modals. The effectiveness of the proposed method is verified through numerical examples of spherical SLRSs. Using the method proposed in this paper, the accurate modal expansion results of numerical models can be obtained with a small number of measuring points. The results of the numerical examples indicate that the proposed method can achieve modal expansion through obtained measuring modals and trained GANs without prior knowledge stiffness and mass of SLRS.

Modal Expansion Analysis, Inverse Design, and Experimental Verification of a Broadband High-Aperture Efficiency Circular Short Backfire Antenna Loaded With Anisotropic Impedance Surfaces

In this paper, a broadband highly-efficient circular short backfire antenna (SBFA) loaded with cylindrical anisotropic impedance surfaces (AISs) is proposed. An efficient modal expansion method (MEM) is developed to facilitate the analysis of the AIS-loaded SBFA with a finite-sized flange, which considerably reduces the computational resources as compared to commercial full-wave solvers. By combining the MEM with a multi-objective genetic algorithm, the geometrical dimensions of the antenna and the dispersive properties of the AIS are optimized through an inverse design strategy. As a proof-of-concept example, an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> -band AIS-loaded SBFA with a broad bandwidth and high aperture efficiency is demonstrated, showing a good agreement between measured results and simulation predictions. An aperture efficiency of greater than 83% is experimentally achieved over a broad <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">11</sub> < -10 dB bandwidth of 35.3%, with a peak aperture efficiency of about 100%. Moreover, the underlying operating principle of the antenna is revealed by investigation of the main propagating modes in the SBFA. The extraordinary performance demonstrates that the proposed AIS-loaded SBFA represents a promising candidate for various wireless communication systems.

An Efficient Method for Light Beaming from Subwavelength Slits Surrounded by Surface Gratings

This paper investigates the directional beaming of metallic subwavelength slits surrounded by dielectric gratings. The design of the structure for light beaming was formulated as an optimization problem for the far-field angular transmission. A vertical mode expansion method was developed to solve the diffraction problem, which was then integrated into a genetic algorithm and an active set method to obtain the optimal result. By using the proposed method for a two-slit structure, we demonstrate that both the single- and dual-beaming effects can be efficiently achieved. Moreover, the beaming directions can be flexibly adjusted and precisely controlled.