Modal Expansion Research Articles

An efficient model for the dynamic vehicle-track-bridge-soil interaction system

A semi-analytical approach based on a lumped parameter model is presented for the analysis of the dynamic interaction system of train, track, bridge, and subsoil. Herein, the bridge and the track are modeled as Euler–Bernoulli beams, which are connected through the viscoelastic track bed. The viscoelastic supports of the bridge model capture the flexibility and damping of the subsoil below the foundations. Complex modal expansion of the deformation approximates the response of the non-classically damped bridge-soil subsystem, while a Rayleigh–Ritz approximation is used to efficiently describe the track deflection. To achieve the coupling of the mechanical equations of these subsystems, a variant of component mode synthesis (CMS) is applied. The mass–spring–damper (MSD) system representing the moving train is coupled to the resulting system of equations for the track-bridge-soil subsystem by a discrete substructuring technique (DST). Geometric track irregularities describing the deviation of the track from perfectly straight and smooth are accounted for by random irregularity profile functions. The results of the proposed model are compared with a finite element model to validate the modeling approach. In an application example, the effects of track irregularities and the influence of the track on the dynamic response of a bridge are discussed. The comparison of results with and without soil–structure interaction of the bridge outlines the great influence of the subsoil properties on the dynamic response in case of resonance.

Prediction of vibration responses in a reciprocating compressor interstage piping system using the modal expansion method

Prediction of vibration responses in a reciprocating compressor interstage piping system using the modal expansion method

Optimal sensor placement for parameter estimation and virtual sensing of strains on an offshore wind turbine considering sensor installation cost

This paper proposes an optimal sensor placement (OSP) framework for parameter estimation, virtual sensing, and condition monitoring using information theory. The framework uses a Bayesian OSP method combined with modal expansion to minimize the information entropy about quantities of interest (QoI), such as strain time histories at critical locations of the structure, without the knowledge of input excitation. The proposed optimization framework also accounts for variations in sensor installation cost at different locations on the monitored structure. The framework is evaluated numerically using a realistic model of an offshore wind turbine on a jacket support structure under installation cost assumptions and considering information entropy of the QoI. The QoI in this numerical study are considered to be the strain time history at one or more locations on the support structure in one problem and the parameters of the structure in the other. A correlation length is considered to account for the spatial correlation of data between adjacent sensors. Effects of the correlation length and input loads on the OSP results for parameter estimation are studied. The considered structural parameters for estimation in this study include (1) modulus of elasticity of tower elements (tower stiffness), (2) modulus of elasticity of jacket elements (jacket stiffness), and (3) vertical foundation spring (soil stiffness). The effect of a subjective weight between the information entropy and sensor configuration cost on the OSP results is also investigated. Different optimal designs are achieved for different weight factors, and the Pareto solutions for OSP are presented. It is found that the OSP framework is an effective tool for decision-makers considering the cost of instrumentation. The presented Pareto optimal solutions can give insight into the value of OSP given a limited budget.

Connecting Classical and Quantum Mode Theories for Coupled Lossy Cavity Resonators Using Quasinormal Modes

We present a quantized quasinormal mode approach to rigorously describe coupled lossy resonators and quantify the quantum coupling parameters as a function of distance between the resonators. We also make a direct connection between classical and quantum quasinormal mode parameters and theories, offering new insights into coupled open cavities. We present detailed calculations for coupled microdisk resonators and show striking interference effects that depend on the phase of the quasinormal modes, an effect that is also significant for high-quality-factor modes. Our results demonstrate that commonly adopted master equations for such systems are generally not applicable and we discuss the new physics that is captured using the quantized quasinormal mode-coupling parameters and show how these relate to the classical mode parameters. Using these quasinormal mode insights, we also present several models to fix the failures of the dissipative Jaynes–Cummings-type models for coupled cavities. Additionally, we show how to improve the classical and quantum lossless mode models (i.e., using normal modes) by employing a nondiagonal mode expansion based on the knowledge of the quasinormal mode eigenfrequencies and analytical coupled mode theory to accurately capture the mode interference effects for high-quality factors.

Spatial Domain Indirect Holography for a Full Phaseless Approach to Biomedical Microwave Imaging

Microwave imaging approaches generally require the knowledge of the complex incident field distribution inside the imaging domain. This task can be accomplished by probing the observation domain, thus resulting in a time-consuming process, or assuming a numerical/simulated model. Alternatively, the incident field can be collected on the measurement domain, and the field distribution can be successfully retrieved on the whole imaging space. In this paper, a full phaseless, two-step strategy is described, where the phase of the incident field is retrieved by exploiting a spatial domain indirect holography technique, applied to the acquisition points only. The retrieved complex incident field can be so used for the reconstruction of the incident field inside the domain of interest. More specifically, a modal expansion technique and a source reconstruction method are adopted and compared to perform the above task. Once performing the characterization of the incident field distribution, the proposed strategy is included in the framework of a phaseless inverse scattering problem, where intensity-only data of the total field are involved. The effectiveness of the combined approach is demonstrated by considering an imaging scenario which involves a series of breast models as inhomogeneous targets.

Taxonomy of brane gravity localisations

Generating an effective theory of lower-dimensional gravity on a submanifold within an original higher-dimensional theory can be achieved even if the reduction space is non-compact. Localisation of gravity on such a lower-dimensional worldvolume can be interpreted in a number of ways. The first scenario, Type I, requires a mathematically consistent Kaluza-Klein style truncation down to a theory in the lower dimension, in which case solutions purely within that reduced theory exist. However, that situation is not a genuine localisation of gravity because all such solutions have higher-dimensional source extensions according to the Kaluza-Klein ansatz. Also, there is no meaningful notion of Newton’s constant for such Type I constructions.Types II and III admit coupling to genuinely localised sources in the higher-dimensional theory, with corresponding solutions involving full sets of higher-dimensional modes. Type II puts no specific boundary conditions near the worldvolume aside from regularity away from sources. In a case where the wave equation separated in the non-compact space transverse to the worldvolume admits a normalisable zero mode, the Type III scenario requires boundary conditions near the worldvolume that permit the inclusion of that zero mode in mode expansions for gravitational wave fluctuations or potentials. In such a case, an effective theory of lower-dimensional gravity can emerge at sufficiently large worldvolume distance scales.This taxonomy of brane gravity localisations is developed in detail for linearised perturbations about a background incorporating the vacuum solution of Salam-Sezgin theory when embedded into ten-dimensional supergravity with a hyperbolic non-compact transverse space. Interpretations of the Newton constant for the corresponding Type III localisation are then analysed.

Application and Analysis of Mode Expansion Method in Simulation of Intersatellite Ultra-Long Distance Diffraction Propagation

Application and Analysis of Mode Expansion Method in Simulation of Intersatellite Ultra-Long Distance Diffraction Propagation

New precise results on dynamics of weak-pulse excitation of mirrorless slab of two-level atoms

We consider by precise calculations the result of passing a very short coherent electromagnetic pulse through a gaseous slab of resonant (“two-level”) atoms bounded by vacuum on both sides. By “very short” we mean that at time t=0 (just after the passage) the system is left in an initial condition in which (except in a negligibly small region next to the exit face) the atoms have not yet felt reflection from the exit boundary; we assume only weak excitation from the ground state. A large rôle is played in our calculations by what we call ENR (exact numerical results) which are obtained by summing over a very great number of eigenmodes in the modal expansion of our formulas. Among other things, we show that, for thicknesses exceeding 5 resonant wavelengths λ, the forward flux of radiation is almost precisely inverse to the thickness, contrary to Beer’s law. We also introduce a generalized definition of frequency shift by regarding the forward flux spectrum as a mass distribution and finding the shift in its center of mass. With this definition we are able to exhibit oscillations of extremely small amplitude but extremely well defined period in the shift as a function of thickness, starting around 5λ and losing no quality even at 80λ. We have given a new discussion of the landmark 1969 paper of Burnham and Chiao in which the slowly varying envelope approximation (SVEA) was used to solve the atomic excitation problem exactly subject to neglect of the backward wave. We have elucidated the distinction between the thick-slab approximation (decay of excitation faster than return from back face), used there, and our medium-thickness problem (return from back face faster than local decay). We have shown how the method of Burnham and Chiao can be adapted to the medium-thickness problem, yielding an SVEA result almost identical to the one they presented. Other new results are summarized in the Introduction.

An Analytical Method of Electromagnetic Wave Scattering by a Highly Conductive Sphere in a Lossless Medium with Low-Frequency Dipolar Excitation

The current research involves an analytical method of electromagnetic wave scattering by an impenetrable spherical object, which is immerged in an otherwise lossless environment. The highly conducting body is excited by an arbitrarily orientated time-harmonic magnetic dipole that is located at a reasonable remote distance from the sphere and operates at low frequencies for the physical situation under consideration, wherein the wavelength is much bigger than the size of the object. Upon this assumption, the scattering problem is formulated according to expansions of the implicated magnetic and electric fields in terms of positive integer powers of the wave number of the medium, which is linearly associated to the implied frequency. The static Rayleigh zeroth-order case and the initial three dynamic terms provide an excellent approximation for the obtained solution, while terms of higher orders are of minor significance and are neglected, since we work at the low-frequency regime. To this end, Maxwell’s equations reduce to a finite set of interrelated elliptic partial differential equations, each one accompanied by the perfectly electrically conducting boundary conditions on the metal sphere and the necessary limiting behavior as we move towards theoretical infinity, which is in practice very far from the observation domain. The presented analytical technique is based on the introduction of a suitable spherical coordinated system and yields compact fashioned three-dimensional solutions for the scattered components in view of infinite series expansions of spherical harmonic modes. In order to secure the validity and demonstrate the efficiency of this analytical approach, we invoke an example of reducing already known results from the literature to our complete isotropic case.

Higher mode effects on capacity curves

The effects of higher modes on the capacity curves and their effects on the formation of plastic hinges are studied herein for regular reinforced concrete structure with no any weak or soft story having time period typically less than 1 sec. The theoretical validity of modal pushover analysis is explained with respect to elastic and inelastic systems. In this modal pushover analysis (MPA), the seismic demand due to individual terms in the modal expansion of the effective earthquake forces is determined by a pushover analysis using the inertia force distribution for each mode. The higher mode pushover curves show significant increase in base shear as well as significant decrease in roof displacement that are not detected by the first mode which is a part of FEMA-356 force distributions. To determine the difference of seismic response parameter at concerned point, local performance levels are used rather than global performance level for better accuracy. Fiber plastic hinges model is used instead of conventional lumped plastic hinge where stress-strain curves is directly applied for confined concrete to inner core concrete and unconfined stress- strain to outer concrete. Base Shear of Capacity Curve at IO performance level, inelastic beam component with compressive stress at 0.002 and 0.0038 increased by 14.52 %, 10.825 %, and 10.42 % respectively. Displacement Demand of Capacity Curve at IO performance level, inelastic beam component with compressive stress at 0.002 and 0.0038 decreased by 15.84 %, 15. 94 % and 12.86 %. It is seen that up to 3rd mode is crucial and should be amalgamate in regular structure and up to 8th mode should be amalgamate in irregular structure

Effect of geometric imperfections and circumferential symmetry on the internal resonances of cylindrical shells

In the present work the resonant response of an imperfect cylindrical shell is investigated using Donnell’s nonlinear shallow shell theory. For this, a reduced order multi degree of freedom model is obtained by discretizing the partial differential equation of motion using the Galerkin procedure and a consistent modal expansion which captures the nonlinear modal interactions and couplings between two asymmetric vibration modes with near commensurable natural frequencies in a 1:2 ratio. As a result of the circumferential symmetry each mode exhibits a 1:1 internal resonance, leading to a possible 1:1:2:2 multiple internal resonances. These modes are coupled through quadratic and cubic nonlinearities arising from the shell curvature and nonlinear strain–displacement relation. The existence and stability of solutions and their bifurcations are investigated using numerical continuation methods for bifurcation analysis and their stability are studied using Floquet theory. It is known that geometric imperfections have a strong influence on the response of thin shell structures. Here, a detailed parametric analysis shows the influence of different forms of geometric imperfections on the shell natural frequencies and bifurcations in the main resonance region. Several branches of solutions due to multiple bifurcations are detected leading to dynamic jumps under increasing and decreasing frequency sweep. Steady-state harmonic and quasi-periodic responses resulting from Neimark–Sacker bifurcations are detected. The reduced order model demonstrates the influence of the geometric imperfection shape and magnitude on the bifurcation scenario and the energy transfer among the four interacting modes.

Modal approximation for strictly convex plasmonic resonators in the time domain: The Maxwell's equations

We study the possible expansion of the electromagnetic field scattered by a strictly convex metallic nanoparticle with dispersive material parameters placed in a homogeneous medium in a low-frequency regime as a sum of modes oscillating at complex frequencies (diverging at infinity), known in the physics literature as the quasi-normal modes expansion. We show that such an expansion is valid in the static regime and that we can approximate the electric field with a finite number of modes. We then use perturbative spectral theory to show the existence, in a certain regime, of plasmonic resonances as poles of the resolvent for Maxwell's equations with non-zero frequency. We show that, in the time domain, the electric field can be written as a sum of modes oscillating at complex frequencies. We introduce renormalised quantities that do not diverge exponentially at infinity.

Livestock, Colonialism, and Commodity Frontiers in the U.S. Southwest

Commodity frontiers are a useful way to think about the expansion and rearticulation of capitalist modes of production across the globe. A weakness of this approach is to miss deeper histories of colonialism and domination at the sites of the metaphorical frontier. This commentary thinks through Diné relationship with sheep to think through how livestock often contains older relationships that transcend colonial limitations.

Hydroelastic analysis of floating structures under wave loads

Accurate prediction of hydroelastic response in ocean waves is of great significance to the structural design and reliability design of floating structures. In this paper, based on the potential flow theory, a large floating structure is simplified as a thin-plate material, and the hydrodynamic characteristics of the structure are calculated by using the modal expansion method and the boundary element method. The correctness of the theory and calculation is verified by comparing the experimental and numerical results. Further, the wave properties and structural materials characterization were changed, this paper calculates the stress and deflection of the structure under wave action, and analyzes the effects of hydroelastic response on the safety of the structure.

Effect of a perfectly conducting corner space on the PAFs for an absorptive dielectric circular cylinder

The photophoretic asymmetry factors (PAFs) for an absorptive dielectric circular cylinder, located near a perfectly conducting and totally reflecting corner space are derived and computed. The method used in this analysis relies on the modal expansion method in cylindrical coordinates, the classical method of images, and the translational addition theorem of cylindrical wave functions. Initially, the components of the internal electric field vector are obtained stemming from an analysis of the scattering. Subsequently, the solution is used to integrate to the normalized intensity function over the cylinder’s volume to obtain the longitudinal (L) and transverse (T) PAFs. Both TM- and TE-polarized plane progressive waves with arbitrary incidence (in the polar plane) are considered. Attention is given to varying the dimensionless size parameter of the cylinder, the angle of incidence of the incoming waves, and the dimensionless distance parameters from the corner space. Numerical examples illustrate the analysis and demonstrate the net effect of the totally reflecting corner space on the L- and T-PAFs, where negative, positive, and neutral values have been predicted. The results are relevant in applications related to the emergence of the photophoretic force and torque on an absorptive particle located near surfaces and topics in electromagnetic/optical scattering, particle manipulation and assembly, optically bound matter, light–matter interactions, and photopheresis.

Structure-acoustic simulation using the modal expansion method and the optimum sensor placement

Structure-acoustic simulation using the modal expansion method and the optimum sensor placement

Fatigue Stress Estimation for Submerged and Sub-Soil Welds of Offshore Wind Turbines on Monopiles Using Modal Expansion

The design of monopile foundations for offshore wind turbines is most often driven by fatigue. With the foundation price contributing to the total price of a turbine structure by more than 30%, wind farm operators seek to gain knowledge about the amount of consumed fatigue. Monitoring concepts are developed to uncover structural reserves coming from conservative designs in order to prolong the lifetime of a turbine. Amongst promising concepts is a wide array of methods using in-situ measurement data and extrapolating these results to desired locations below water surface and even seabed using models. The modal decomposition algorithm is used for this purpose. The algorithm obtains modal amplitudes from acceleration and strain measurements. In the subsequent expansion step these amplitudes are expanded to virtual measurements at arbitrary locations. The algorithm uses a reduced order model that can be obtained from either a FE model or measurements. In this work, operational modal analysis is applied to obtain the required stress and deflection shapes for optimal validation of the method. Furthermore, the measurements that are used as input for the algorithms are constrained to measurements from the dry part of the substructure. However, with subsoil measurement data available from a dedicated campaign, even validation for locations below mud-line is possible. After reconstructing strain history in arbitrary locations on the substructure, fatigue assessment over various environmental and operational conditions is carried out. The technique is found capable of estimating fatigue with high precision for locations above and below seabed.

Advances in theoretical, experimental and applied studies on second-harmonic generation of ultrasonic Lamb waves

<p indent=0mm>Nondestructive evaluation and testing of material degradation or micro-damages in an early stage are essential for the maintenance of the safety-critical structures. Therefore, it is important to detect defects in a very early stage or even better to detect changes in the microstructure before the defect initiation even occurs. Linear ultrasonic characteristics are not sufficiently sensitive to the kind of microscopic degradation of the material. However, on the other hand, even very small imperfections can produce very significant excess nonlinearity, which can be orders of magnitude higher than the intrinsic nonlinearity of the intact material. The excess nonlinearity is produced mainly by the strong local nonlinearity of microcracks. In order to assess the damage in components at an early stage, the use of nonlinear acoustics has been shown to be the leading and one of the most promising techniques. Second harmonic generation is a typical acoustic nonlinear effect of ultrasonic waves, which can be used to assess the material degradation and detect the micro-defects in their early stages. Furthermore, ultrasonic Lamb wave has been widely used in large-scale industrial structures for health monitoring and material nondestructive testing. Possibility to inspect inaccessible or hidden areas and great cost-effectiveness are the key advantages of ultrasonic Lamb wave-based damage detection approaches. Second harmonic Lamb wave, which combines the advantages of Lamb wave and the high sensitivity of second harmonic waves to micro damage, is drawing increasing attention in recent years. The measurement of second harmonic Lamb waves is investigated as a promising feature extraction technique to characterize material nonlinearity in plate-like structures. However, the second harmonic fields of Lamb waves are much more complex due to the dispersion and multi-mode nature of propagating Lamb waves. In general, the effect of second harmonic generation is often very small and can easily be overlooked experimentally. Consequently, proper mode tuning with physically based feature is essential to enhance the efficiency of second harmonic Lamb wave generation and reception. In experimental work, it is also difficult to observe and measure the second harmonic Lamb waves. Although significant development has been made in theoretical analysis and experimental investigation of second Lamb waves, the use of second harmonic Lamb wave for structural damage assessments is still limited by a series of scientific and technical difficulties. In this paper, the theoretical and experimental advances concerning second harmonic Lamb wave are systematically reviewed. First, theoretical analysis of second harmonic Lamb waves based on nonlinear reflection of acoustic waves is reviewed. Then, a general approach of perturbation method and normal modal expansion for theoretical study of the second harmonic Lamb waves propagation in a plate is introduced. The physical insight of nonlinear feature among different Lamb modes is clarified. As revealed in these pieces of theoretical work, the second harmonic modes of Lamb waves display an accumulative effect when conditions of phase matching and non-zero power transfer from the fundamental mode to the second harmonic mode are satisfied. The conditions for strong nonlinear effect of Lamb wave are discussed. Experimental observation and measurement of second harmonic Lamb waves are reviewed. Cumulative effect and efficiency of second harmonic generation of Lamb waves are discussed. The guideline for choosing the Lamb modes with higher sensitivity to material nonlinearity is provided. Based on the theoretical and experimental studies, the applications of second harmonic Lamb waves for material damage evaluation are summarized. In addition, some obscure understanding and disputed issues during the study in this research area are clarified. Finally, prospective priorities and research trends of second harmonic Lamb waves are discussed.

Experimental Dispersion identification using a fitted state-space model

An identification method that can estimate the dispersion relation of waveguides experimentally using an efficient and accurate procedure is presented. The method fits a linear state-space model before resorting to a kinematic wave model in the frequency region near the pre-identified natural frequencies. The eigenvectors, or mode-shapes, are computed at the sensor locations, and based on the reduced-Bloch mode expansion method, the propagation modes are fitted to match the identified vibration mode-shape at these frequencies. Classical methods to identify the dispersion relation from measured data can be computationally expensive and time-consuming, with limited accuracy in cases of multimode propagation. Results show that the fitted dynamic model expands the frequency range of obtained dispersion curves and enhances the speed of computation and accuracy. The method is derived and verified for both lumped and distributed systems, using numerical finite element simulation. Experimental verification is carried out on two acoustical waveguides. The first is a circular ring-shaped array of coupled Helmholtz resonators, modeled using a lumped parameter model. The second is an air-filled acoustic wave-tube that is modeled as a distributed acoustic-elastic coupled waveguide. The method's strengths and weaknesses are discussed from the experimentally obtained dispersion curves, and its main feature, the ability to fit the dispersion model of weak modes, is highlighted.

Wave energy extraction and hydroelastic response reduction of modular floating breakwaters as array wave energy converters integrated into a very large floating structure

Combing floating breakwaters with wave energy converters (WECs) and integrating them into very large floating structure (VLFS) can provide a viable option to explore economically offshore wave energy resources and simultaneously to protect marine structures. In this paper, the time-domain numerical model is developed based on the modal expansion theory with nonlinear consideration to optimize the design and layout of an integrated system of modular WEC-type floating breakwaters and a pontoon-type VLFS, with emphasis on the effects of the WEC geometric size and shape, the WEC-VLFS gap distance and the wave nonlinearity. A hybrid finite element (FE)-boundary element (BE) method is presented to simulate the structures as Mindlin plate elements and the water waves as fully nonlinear potential flow boundaries, respectively. Breakwaters as WECs with deeper draft and larger length are found to more fully interact in phase with long-period waves, and receive more wave energy extraction and larger hydroelastic response reduction. The addition of breakwaters has a favorable effect on the wave energy extraction, but a destructive effect on the hydroelastic reduction. Importantly, wave resonance induced by the multi-modal scattering waves in the WEC-VLFS gap leads to multiple peaks of the power capture efficiency. Compared to the symmetric-shape WECs, the asymmetric-shape WECs strengthen the gap resonant effect, which improves both the wave energy extraction and hydroelastic reduction for a broader frequency bandwidth. The findings of this study indicate the synergistic benefits of wave energy exploitation and transmitted wave attenuation at the fore-end of VLFSs.