Normal Mode Expansion Research Articles

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.

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.

Analytical insight into local defect resonance using a two-step method

Local defect resonance (LDR) has gained increasing attention in the field of non-destructive testing due to its capacity to selectively amplify vibrations within the defect regions and to enable defect-selective and high-resolution imaging. How-ever, the conventional theory based on vibration method adopted for analyzing the fundamental frequencies of LDR deviates from experimental results when dealing with relatively thick defects. To tackle this deficiency, this study presents a two-step analytical methodology designed to precisely evaluate the local reso-nance effects of thick planar defects. In the first step, the normal-mode expansion method is employed to calculate the phase shift of the reflected elastic wave. In the second step, by using the standing wave's wavelength obtained in the first step, the Rayleigh method is taken into account to calculate LDR frequencies for various defect shapes. The refined theory is subsequently validated through finite element analysis and experimental measurement. The results clearly demonstrate that the LDR frequencies calculated using the refined theory exhibit a better agreement with experimental and simulated data.

Generation of quasi-traveling waves in a finite rectangular membrane with two internal viscoelastic line supports

In this work we study the forced vibration of a finite rectangular membrane driven by arbitrarily distributed boundary motion on two adjacent edges and internally coupled with multiple viscoelastic line supports, as an extension of the analysis of coexisting vibrations and waves in one-dimensional elastic strings and ducts. Using the static solution for the membrane displacement in the absence of interior supports, the response of the nonclassically damped system is obtained through a normal-mode expansion. A traveling-wave index based on complex orthogonal decomposition is utilized to identify and optimize the traveling waves in the rectangular membrane. A vibration localization condition in a semi-infinite membrane is analytically derived, from which the complex boundary excitations and constraints in a finite membrane are specified to effectively realize pure traveling waves and confine the vibration to a restricted area. Then the membrane is edge-driven by a simpler time-periodic, spatially sinusoidal excitation for practical realization, with the line-support parameters optimized using the traveling-wave index function and a derivative-free optimization routine with lower-bound constraints. Analytical results show that quasi-traveling waves are produced for ranges of line-support parameters (location, stiffness and damping) following optimization using the traveling-wave index. The developed wave patterns are examined with a power-flow analysis and numerically verified through finite element analysis. Wave localization is demonstrated to be valid in a finite membrane with specific boundary conditions and multiple viscoelastic line supports. The proposed method shows the predictive ability of the orthogonally stiffened membrane configuration as the basic unit cell for the design of periodic structures with directional wave propagation as well as vibration localization dynamics.

Behaviour of circumferential and helical guided waves in two-layered composite cylindrical shell

A model of the acoustic scattering of a plane wave incident obliquely upon an infinite elastic bi-layered cylindrical shell is developed based on normal mode expansion technique. The far field backscattering form function for the composite copper/stainless steel cylindrical shell is calculated for reduced frequency range of k⊥a1=0–70 and for different values of incident angles α (k⊥=k1cos⁡α, k1 is the wave number in the fluid surrounding the shell and a1 is the outer radius of the shell). Numerical calculations show the behaviour of the resonances of circumferential and helical guided waves as the angle of incidence increases. The obtained results are compared with those of copper and stainless steel one-layered cylindrical shells. The use of frequency-angle representation depicts the helical propagation of surface waves along the axis of the shell. This representation, along with the identification plan, enables to describe the evolution of acoustic scattering as a function of incidence angle and vibration mode n. Furthermore, a comparative analysis of the resonance trajectory evolution between mono-layered and composite cylindrical shells is performed. The evolution of the helical guided waves, the bifurcation of the A0 wave into the A0− and A0+ waves, and a region with high backscattering are investigated. The presented results show that the extension of the region with high backscattering depends on the thickness and physical properties of the bi-layered cylindrical shell.

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.

Reducing the Cost of Neural Network Potential Generation for Reactive Molecular Systems.

Although machine learning potentials have recently had a substantial impact on molecular simulations, the construction of a robust training set can still become a limiting factor, especially due to the requirement of a reference ab initio simulation that covers all the relevant geometries of the system. Recognizing that this can be prohibitive for certain systems, we develop the method of transition tube sampling that mitigates the computational cost of training set and model generation. In this approach, we generate classical or quantum thermal geometries around a transition path describing a conformational change or a chemical reaction using only a sparse set of local normal mode expansions along this path and select from these geometries by an active learning protocol. This yields a training set with geometries that characterize the whole transition without the need for a costly reference trajectory. The performance of the method is evaluated on different molecular systems with the complexity of the potential energy landscape increasing from a single minimum to a double proton-transfer reaction with high barriers. Our results show that the method leads to training sets that give rise to models applicable in classical and path integral simulations alike that are on par with those based directly on ab initio calculations while providing the computational speedup we have come to expect from machine learning potentials.

Discrepant Approaches to Modeling Stellar Tides and the Blurring of Pseudosynchronization

We examine the reasons for discrepancies between two alternative approaches to modeling small-amplitude tides in binary systems. The direct solution (DS) approach solves the governing differential equations and boundary conditions directly, while the modal decomposition (MD) approach relies on a normal-mode expansion. Applied to a model for the primary star in the heartbeat system KOI-54, the two approaches predict quite different behavior of the secular tidal torque. The MD approach exhibits the pseudosynchronization phenomenon, where the torque due to the equilibrium tide changes sign at a single, well-defined, and theoretically predicted stellar rotation rate. The DS approach instead shows “blurred” pseudosynchronization, where positive and negative torques intermingle over a range of rotation rates. We trace a major source of these differences to an incorrect damping coefficient in the profile functions describing the frequency dependence of the MD expansion coefficients. With this error corrected, some differences between the approaches remain; however, both are in agreement that pseudosynchronization is blurred in the KOI-54 system. Our findings generalize to any type of star for which the tidal damping depends explicitly or implicitly on the forcing frequency.

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.

Reducing ambiguities in single-hydrophone matched-field processing by exploiting source motion

Matched-field processing for localizing an underwater acoustic source in range and depth from power-spectrum measurements obtained at a single hydrophone receiver often suffers from high side-lobes and ambiguities when the signal is narrowband or signal bandwidth is small. In this paper we review how source motion can be utilized to reduce side-lobes in depth and range via an incoherent synthetic-aperture-like approach and explain the mechanisms responsible for side-lobe reduction relative to the height of the main-lobe by using a normal-mode expansion for the pressure field. We also derive an approximation for the depth main-lobe width when the true source range is known for the ideal rigid-waveguide case. Numerical results are presented corroborating the analytical analysis along with some matched-field localization ambiguity surface examples for (a) an ideal shallow-water waveguide with a pressure-release top boundary and a rigid bottom boundary and (b) a more realistic shallow-water Pekeris environment, to demonstrate how side-lobes and ambiguities are reduced when source motion is exploited in the matched-field processing.

The Effect of Linear, Parabolic and Inverted Parabolic Salinity Gradients on The Onset of Darcy Brinkman Rayleigh Benard Two-Component Convection in a Two-Layered System with Dufour Effect

The physical configuration of the problem of Darcy Brinkman (DB) Rayleigh Benard Two-Component (RBTC) Convection in a two-layered system has been investigated for linear, parabolic and inverted parabolic salinity gradients with the Dufour effect. For the fluid layer, the upper boundary is free with surface tension and for the porous layer, the lower boundary is rigid. At the interface, the normal velocity, normal stress, shear stress, mass, mass flux, heat, and heat flux are continuous. The regular perturbation method is used to solve the resulting ordinary differential equations obtained from normal mode expansion. The effect of different physical parameters on the Rayleigh number versus depth ratio is discussed and results are presented graphically.

Resolution of matched field processing for a single hydrophone in a rigid waveguide.

This paper studies resolution of matched field processing for locating, in range and depth, a broadband underwater acoustic source from data measured at a single hydrophone receiver. For the case of an ideal rigid shallow-water waveguide with a pressure-release top boundary and a rigid bottom boundary, the paper derives approximations for the main-lobe widths of the ambiguity surface. The two cases studied in this paper are (1) when coherent measurements of the pressure are available, with the transmitted source waveform precisely known, and (2) when only measurements of the received-signal pressure magnitude-squared are available, such as might occur when the transmitted signal is random and unknown. The analysis uses the normal-mode expansion for the pressure field to derive approximate expressions for the ambiguity-surface main-lobe widths, as a function of the number of modes and frequency band, for both range and depth. Numerical results are presented corroborating the analytical analysis. Finally, the paper argues that this ambiguity analysis can also give insights into Pekeris waveguides under appropriate conditions and shows numerical simulations of matched-field localization ambiguity surfaces for some realistic shallow-water Pekeris environments.

Internal damage localization for large-scale hollow cylinders based on helical sensor network

The hollow cylinder with a large diameter is the typical shell structure in major engineering applications such as aviation, aerospace, ocean, and so on. Unlike the geometric properties of transportation pipe, the shell structure has a large size and short length. It is relatively difficult to detect damage globally. According to the abovementioned characteristics, a damage localization method based on helical sensor network is proposed in the current research work to locate the internal damage in large-scale hollow cylinder in which the sensors are arranged on the outer surface of the structure. The wavefront theory about flexural modes is fully explored to determine helical sensor network which is suitable for the full frequency spectrum and avoids the approximate estimation to the wave in plate. The flexible sensing paths and single mode excitation in helical sensor network also provide more possibilities for the damage localization in hollow cylinder. The verification method of circumferential orders based on the Fourier transform is developed by the normal mode expansion approach. To apply the helical sensor network of flexural mode in damage localization, the effective loading length of flexural modes is investigated using simulation. And the simulation and experiment both are conducted to locate the internal damage in hollow cylinders and verify the credibility of proposed localization method. The scale parameter and sensing path number are further investigated to determine the effect on location accuracy.

Physically agnostic quasi normal mode expansion in time dispersive structures: From mechanical vibrations to nanophotonic resonances

Resonances, also known as quasi normal modes (QNM) in the non-Hermitian case, play an ubiquitous role in all domains of physics ruled by wave phenomena, notably in continuum mechanics, acoustics, electrodynamics, and quantum theory. In this paper, we present a QNM expansion for dispersive systems, recently applied to photonics but based on sixty year old techniques in mechanics. The resulting numerical algorithm appears to be physically agnostic, that is independent of the considered physical problem and can therefore be implemented as a mere toolbox in a nonlinear eigenvalue computation library.

Wide frequency band expansion of permittivity normal modes

Normal modes are valuable tools for modeling electromagnetic resonators, since all their electromagnetic properties can be extracted from a small set of modes. To extend the utility of normal modes to open systems, a set of modes was developed where permittivity is designated to be the eigenvalue. However, these modes, also known as generalized normal modes, are defined at only a single frequency, which limits their utility for spectral applications. In this paper, we present a simple way to extend the validity of permittivity modes to neighboring frequencies. This enables the evaluation of spectral lineshapes and scattering of short pulses from open nanophotonic structures using knowledge of the generalized normal modes at only a single frequency.

Scattering of guided waves propagating through pipe bends based on normal mode expansion

The scattering of guided waves propagating through pipe bends is studied by means of normal mode expansion. First, the bi-orthogonality relationship for normal modes in pipe bends is derived, based on which the displacement and stress fields at the interfaces between the straight and curved parts are expanded with the normal modes in both parts. Then, based on the displacement and stress field continuity principle, the scattering problem is regarded as an eigenproblem of a transfer matrix, the solution of which gives the mode conversions at the interfaces. A case study is presented of the low-frequency longitudinal mode incident on a pipe bend, and it is found that the dominant mode conversions are L(0,1) reflection and mode conversion from L(0,1) to F(1,1). Finite element simulations and experiments are also conducted. L(0,1) bend reflection and mode-converted F(1,1) are clearly observed, which agrees well with the theoretical predictions.

Generalized normal mode expansion method for open and lossy periodic structures

We describe and demonstrate the extension of permittivity mode expansion, which is also know as generalized normal mode expansion (GENOME), to open and lossy periodic structures. The resulting expansion gives a complete spatial characterization of any open periodic structure, via the quasi-periodic Green’s tensor, by a complete, discrete set of modes rather than a continuum. The method has been validated by comparing our expansion of an open waveguide array with a direct scattering calculation. Good agreement was obtained regardless of the source location or detuning from resonance.

Resolution of matched field processing for a single hydrophone in a rigid waveguide

This paper studies resolution of matched field processing for locating, in range and depth, a broadband underwater acoustic source from data measured at a single hydrophone receiver. For the case of an ideal rigid shallow-water waveguide with a pressure-release top boundary and a rigid bottom boundary, we derive approximations for the main-lobe widths of the ambiguity surface. The two cases studied in this paper are (1) when coherent measurements of the pressure are available, with the transmitted source waveform precisely known, and (2) when only measurements of the received signal power spectral density (PSD) are available, such as occur when the transmitted signal is random and unknown. The analysis uses the normal-mode expansion for the pressure field to derive approximate expressions for the ambiguity surface main-lobe widths, as a function of the number of modes and frequency band, for both range and depth. Numerical results are presented corroborating the analytical analysis. Finally, we argue that this ambiguity analysis also gives insights into real ocean waveguide localization characteristics under appropriate conditions, and show numerical simulations of matched field localization ambiguity surfaces for some realistic shallow-water Pekeris environments.

Multi-Mode Ultrasonic Guided Waves Based Damage Detection in L-Bars with Asymmetric Cross-Section with Sum of Multiple Signals Method

Bars are significant load-carrying components in engineering structures. In particular, L-bars are typical structural components commonly used in truss structures and have typical irregular asymmetric cross-sections. To ensure the safety of load-carrying bars, much research has been done for non-destructive testing (NDT). Ultrasonic guided waves have been widely applied in various NDT techniques for bars as a result of the long-range propagation, low attenuation, and high sensitivity to damages. Though good for inspection of ultrasonic guided waves in symmetric cross-section bar-like structures, the application in asymmetric ones lacks further research. Moreover, traditional damage detection in bars using ultrasonic guided waves usually depends on a single-mode at a lower frequency with lower sensitivity and accuracy. To make full use of all frequencies and modes, a multi-mode characteristic-based damage detection method is presented with the sum of multiple signals (SoM) strategy for L-bars with asymmetric cross-section. To control the desired mode in multi-mode ultrasonic guided waves, excitation optimization and weighted gathering are carried out by the analysis of the semi-analytical finite element (SAFE) method and the normal mode expansion (NME) method. An L-bar example with the asymmetric cross-section of 35 mm × 20 mm × 3 mm is used to specialize the proposed method, and some finite element (FE) models have been simulated to validate the mode control. In addition, one PZT is applied as a contrast in order to validate the multielement mode control. Then, more FE simulations experiments for damage detection have been performed to validate the damage detection method and verify the improvement in detection accuracy and damage sensitivity.

Developing efficient finite element techniques for atmospheric sound propagation

Applying the finite element method to problems in atmospheric sound propagation is challenging because the use of a standard approach, such as meshing the entire domain, quickly delivers a prohibitive problem size. The finite element method does, however, offer many advantages such as the ability to accommodate continuous variations in fluid properties, as well as scattering from obstacles of complex geometry. This means the method is ideally placed to provide benchmark solutions for more popular approaches. Accordingly, ideas for developing more efficient versions of the finite element method suitable for atmospheric sound propagation are explored here for two dimensional problems. These are based on the use of one dimensional normal mode expansions, and then mapping these on to two dimensional solutions for non-uniform obstacles. The normal mode solution for a range independent inhomogenous moving fluid is presented, and numerical mode matching techniques are introduced to demonstrate the inclusion of a point source. The extension of these mode matching methods to accommodate scattering from obstacles is then discussed, and the conditions necessary to deliver efficient finite element solutions are reviewed.