Superposition Of Normal Modes Research Articles

Modern Methods of Sound Propagation Modeling Based on the Expansion of Acoustic Fields over Normal Modes

A review of modern methods of modeling acoustic fields based on their representation as a superposition of normal modes is presented. Most of the described methods are based on an approach to calculating mode amplitudes by solving parabolic equations of various types, both narrow-angle and wide-angle. We also consider two-dimensional methods for calculating acoustic fields, to which the above-mentioned three-dimensional approaches are reduced in the absence of dependence of the field and medium parameters on one of the horizontal coordinates. The computation of both time-harmonic acoustic fields and pulsed sound signals is discussed. A number of numerical examples are considered in which such calculations are performed taking into account three-dimensional sound propagation effects. For the first time within the framework of this approach, the calculation of particle accelerations at the pulse signal reception points, as well as the calculation of the energy density flux of the vector field were performed.

Normal mode energy estimation based on reconstructing the incoherent beamformed outputs from a horizontal array.

The acoustic pressure field in many underwater environments is well described by a superposition of normal modes. The normal modes can be used for source localization and environmental inversion. However, the wavenumber resolution of traditional normal mode filtering methods for a small-aperture horizontal array is usually not sufficient to identify individual modes in a shallow water waveguide. This paper proposes an original method of normal mode energy estimation to remove the energy leakage between modes. The modal energy is defined as the square of the modal amplitude. This method is to reconstruct the incoherent beamformed outputs in wavenumber domain for a horizontally moving source. The adaptive beamforming is used to suppress interference and improve output signal-to-noise ratio. The uncertainty of modal phase velocity has also been considered in this method. The proposed method can provide more accurate estimates of modal energy for a small-aperture horizontal array than the traditional mode filtering methods, such as the matched filter, the least squares mode filter, the regularized-least squares mode filter, and the maximum a posteriori mode filter, in simulations and experiments.

Combining empirical wavelet transform and transfer matrix or modal superposition to reconstruct responses of structures subject to typical excitations

An investigation is presented in this paper into the applicability of empirical wavelet transform, combined with mode transfer matrix and/or modal superposition, to reconstruction of responses of structures subject to typical excitations. Empirical wavelet transform, of which the parameters are determined from scale-space representation, is used to decompose each measured response signal into a number of mono-components. Structural responses at unmeasured locations are reconstructed from those mono-components, either through superposition of normal modes, or through mode transfer matrices, both derived from a finite element model of the structure in question. The method is applied to a high-speed train-used extruded aluminium panel subject to typical excitations. Considered excitations include impact excitations, stochastic excitations, periodic excitations and periodic impact excitations. The accuracy of the method is demonstrated both numerically and experimentally. The robustness of the method to measurement noise is also investigated for the panel. Results show that the proposed method is not only able to overcome difficulties in existing methods caused by shortages in measurement point, mode mixing and/or periodic excitations, but also robust to measurement noise.

Skeletal animation for visualizing dynamic shapes of macromolecules

AbstractWe apply a skeletal animation technique developed for general computer graphics animation to display the dynamic shape of protein molecules. Polygon-based models for macromolecules such as atomic representations, surface models, and protein ribbon models are deformed by the motion of skeletal bones that provide coarse-grained descriptions of detailed computer graphics models. Using the animation software Blender, we developed methods to generate the skeletal bones for molecules. Our example of the superposition of normal modes demonstrates the thermal fluctuating motion obtained from normal mode analysis. The method is also applied to display the motions of protein molecules using trajectory coordinates of a molecular dynamics simulation. We found that a standard motion capture file was practical and useful for describing the motion of the molecule using available computer graphics tools.

Comparative study of short-term extreme responses and fatigue damages of a floating wind turbine using two different blade models

In this work, two different blade structural models are used to estimate the blade deformations and the global structural responses of a 10MW floating offshore wind turbine (FOWT). One model is based on the Euler-Bernoulli beam theory and it is solved by the linear normal mode superposition method. The other model is based on the geometry exact beam theory (GEBT) which can consider the full geometric nonlinearity and large deformation. The control equations of GEBT are discretized by Legendre spectral finite elements. The aero-hydro-servo-elastic fully coupled numerical simulations are conducted in the open-source analysis tool OpenFAST to explore the feasibility of the two different structural models for modeling large scale wind turbine blades. Both the steady-state and dynamic results show that power generation and thrust on rotor are similar for the different blade models. There is a small difference in the results of the blade pitch angle and flapwise and edgewise blade root bending moment at high wind speeds due to the lack of torsion degree of freedom in the mode-based method. The difference between the two models is mainly reflected in the prediction of blade tip deformations. The one-hour short-term extreme blade root bending moments and the damage equivalent fatigue loads at blade root are both compared based on the two models. For edgewise bending moment, the extreme value of GEBT model is found at cut-out wind speed, whereas the linear beam model predicts the extreme value around rated wind speed. For the flapwise bending moment, the extreme value is captured around the rated wind speed for both of the two models, but GEBT model presents a larger value. As for fatigue loads, the short-term 1 Hz damage equivalent loads calculated based on the linear beam model are smaller than GEBT model at almost all load cases for both edgewise and flapwise root bending moment, which implies that the linear beam model may underestimate the life time fatigue damage at blade root.

Detailed prediction of fluid-solid coupled phenomena of turbulent flow through reed valves

The operation of self-actuated valves in suction and discharge processes plays a crucial role in the thermal efficiency and structural reliability of hermetic reciprocating compressors. The present work provides a thorough description of an in-house numerical method based on a partitioned fluid-structure interaction algorithm intended to obtain high-fidelity numerical predictions and optimise valve system design. A large eddy simulation model is used for the turbulent flow, whereas a normal mode superposition assumption along with an impact penalty method is used for the reed. Under this framework, a detailed analysis of the coupled problem with a straight and tapered exit ports is carried out. Differences in effective flow area and pressure drop are found as a result of the port geometry, but also of the valve velocity.

Improved algorithms applying the numerical Laplace method for response analyses of Timoshenko beam subjected to typical external loads

This study develops Su and Ma’s theory regarding the stationary load case by considering the structural damping (which can be arbitrarily proportional or non-proportional) and various boundary constraints. Original derivations are presented based on Timoshenko’s beam model and the Kelvin–Voigt damping model. Numerical Laplace inversion, i.e., the Durbin method, is applied to obtain the time-domain solutions. Further, an extension to the general moving load case is performed by means of the finite element method, and an algorithm by virtue of mathematical fast transformation is exploited to improve the computational efficiency. The normal mode superposition method is introduced to validate the numerical solutions. In case studies, the accuracy of the Laplace method is first verified through contrastive studies. Based on that analysis, numerical experiments regarding the moving load case are conducted while considering the implications of moving velocities and non-proportional damping. The results demonstrate that the numerical method is effective and efficient for typical boundary conditions, although this approach suffers from certain algorithm-based instabilities. The load velocity and the structural damping are fundamental influential factors on the dynamical performance, in which even divergence appears in a damped system, and regular piecewise-linear rules are concluded in non-proportional damping variations. Since damping is a fundamental parameter for structural dynamics and the stationary and moving loads consist of the basic excitations, the Laplace method has a strong application prospect.

Transient Beam Response due to Point-Load Impact and Estimation of Onset of Yielding in Bending

When a dynamic point load is applied laterally to a beam, the deflection gradually spreads over time until the entire beam length is affected. The determination of the early stages of response presents substantial mathematical difficulties. The normal-mode superposition is made awkward by the fact that the number of modal components necessary to get a reasonably accurate answer increases as the time interval of interest decreases. This paper presents a different solution based on a dominating flexural wave, which allows one to formulate compact and transparent expressions.

Kinetic cascade beyond magnetohydrodynamics of solar wind turbulence in two-dimensional hybrid simulations

The nature of solar wind turbulence in the dissipation range at scales much smaller than the large magnetohydrodynamic (MHD) scales remains under debate. Here, a two-dimensional model based on the hybrid code abbreviated as A.I.K.E.F. is presented, which treats massive ions as particles obeying the kinetic Vlasov equation and massless electrons as a neutralizing fluid. Up to a certain wavenumber in the MHD regime, the numerical system is initialized by assuming a superposition of isotropic Alfvén waves with amplitudes that follow the empirically confirmed spectral law of Kolmogorov. Then, turbulence develops and energy cascades into the dispersive spectral range, where also dissipative effects occur. Under typical solar wind conditions, weak turbulence develops as a superposition of normal modes in the kinetic regime. Spectral analysis in the direction parallel to the background magnetic field reveals a cascade of left-handed Alfvén/ion-cyclotron waves up to wave vectors where their resonant absorption sets in, as well as a continuing cascade of right-handed fast-mode and whistler waves. Perpendicular to the background field, a broad turbulent spectrum is found to be built up of fluctuations having a strong compressive component. Ion-Bernstein waves seem to be possible normal modes in this propagation direction for lower driving amplitudes. Also, signatures of short-scale pressure-balanced structures (very oblique slow-mode waves) are found.

Optimization Based Identification of the Dynamic Properties of Linearly Viscoelastic Materials Using Vibrating Beam Technique

Sandwich structures with viscoelastic core and metal face sheets are increasingly used in automotive industry to significantly reduce the amplitude of vibration and noise radiation. Several experimental methods such as dynamic mechanical analysis (DMA) and vibrating beam technique (VBT) are used to characterize the dynamic properties of viscoelastic materials as a function of frequency and temperature. This paper investigates the use of a free-free beam setup, as an alternative solution to the classical clamped-free VBT, for a better control of the effect of boundary conditions on the laminated steel specimen. The new setup is developed in combination with a frequency response function based optimization method, to automatically derive the dynamic properties of viscoelastic core materials and generate their master curves. A solver based on the normal mode superposition method, considering the added mass effect of the impedance head, is used in the cost function of the optimization approach. The sandwich model is based on the Ross–Kerwin–Ungar equation, and the four-parameter fractional derivative model is used in conjunction with the Williams–Landel–Ferry equation to describe the frequency and temperature dependent behavior of the viscoelastic material. The master curves are a direct result of the optimization process. Several applications are described to assess the performance of the present method. In particular, a systematic comparison with both the classical VBT and DMA (when available) is presented.

A Hybrid Method for Determining the Dynamic Stresses of Railway Vehicle

On the basis of the idea that the displacement response of the structure can be expressed as superposition of normal modes of vibration multiplied by corresponding time-dependent modal coordinate, the strain response of the same structure under the same loading condition can also be regarded as a superposition of strain modes multiplied by modal coordinates, this paper presents a hybrid method for determining the dynamic stresses of railway vehicle based on the combination of experiment and numerical simulation. The calculation formulas based on strain mode superposition principle is deduced in detail. The numerical experiment shows that the dynamic stress time history can be obtained by the proposed method expediently and accurately. In addition, to meet the demand of fatigue engineer of the vehicle bogie, a software system was developed, which has a friendly, open and easily removable interface and is valuable for practical use.

Coherent control of stimulated emission inside one dimensional photonic crystals: strong coupling regime

The present paper discusses the stimulated emission, in strong coupling regime, of an atom embedded inside a one dimensional (1D) Photonic Band Gap (PBG) cavity which is pumped by two counter-propagating laser beams. Quantum electrodynamics is applied to model the atom-field interaction, by considering the atom as a two level system, the e.m. field as a superposition of normal modes, the coupling in dipole approximation, and the equations of motion in Wigner-Weisskopf and rotating wave approximations. In addition, the Quasi Normal Mode (QNM) approach for an open cavity is adopted, interpreting the local density of states (LDOS) as the local density of probability to excite one QNM of the cavity; and therefore rendering this LDOS dependent on the phase difference of the two laser beams. In this paper we demonstrate that the strong coupling regime occurs at high values of the LDOS. In accordance with the results of the literature, the emission probability of the atom decays with an oscillatory behaviour, so that the atomic emission spectrum exhibits two peaks (Rabi splitting). The novelty of this work is that the phase difference of the two laser beams can produce a coherent control of both the oscillations for the atomic emission probability and, as a consequence, of the Rabi splitting in the emission spectrum. Possible criteria to design active delay lines are finally discussed.

Transient Response of Circular, Elastic Plates to Point Loads

When a dynamic point load is applied laterally to a plate, the deflection gradually spreads over time until the entire plate surface is affected. The determination of early stages of response presents substantial mathematical difficulties. The normal mode superposition is made awkward by the fact that the number of modal components necessary to get a reasonably accurate answer increases as the time of interest becomes smaller. This brief presents a different approach, based on a flexural wave concept, which solves the problem by formulating compact expressions. At the same time, the approach illustrates the basic mechanism of spreading deflections. A shear correction is introduced for the early phase of motion. The quantification is also of some interest in explaining certain aspects of mass impact against a plate.

Allosteric Communication in Myosin V: From Small Conformational Changes to Large Directed Movements

The rigor to post-rigor transition in myosin, a consequence of ATP binding, plays an essential role in the Lymn–Taylor functional cycle because it results in the dissociation of the actomyosin complex after the powerstroke. On the basis of the X-ray structures of myosin V, we have developed a new normal mode superposition model for the transition path between the two states. Rigid-body motions of the various subdomains and specific residues at the subdomain interfaces are key elements in the transition. The allosteric communication between the nucleotide binding site and the U50/L50 cleft is shown to result from local changes due to ATP binding, which induce large amplitude motions that are encoded in the structure of the protein. The triggering event is the change in the interaction of switch I and the P-loop, which is stabilized by ATP binding. The motion of switch I, which is a relatively rigid element of the U50 subdomain, leads directly to a partial opening of the U50/L50 cleft; the latter is expected to weaken the binding of myosin to actin. The calculated transition path demonstrates the nature of the subdomain coupling and offers an explanation for the mutual exclusion of ATP and actin binding. The mechanism of the uncoupling of the converter from the motor head, an essential part of the transition, is elucidated. The origin of the partial untwisting of the central β-sheet in the rigor to post-rigor transition is described.

Edge resonance in semi-infinite thick pipe: Numerical predictions and measurements

This paper presents theoretical and experimental studies of axisymmetric longitudinal guided wave L(0,2) interaction with the free edge of the pipe. A numerical method based on normal mode superposition is applied to predict the edge resonance by an analysis of dispersion relations of separate modes. In parallel, the finite element analysis and experimental measurements prove the existence of edge resonance in the pipe in case of L(0,2) wave incidence. It is shown that the edge resonance is mainly caused by the first pair of complex modes. Additionally the behavior of edge resonance phenomenon as a function of the curvature of the pipe is studied. The displacement amplitudes measured at the edge demonstrate that the edge resonance is affected by the frequency and thickness to midradius ratio of the pipe, and it is losing its strength in thicker pipes, as the growing difference between the outer and inner radii destroys symmetry. The reflected energy amplitudes show that at the resonance frequencies the incident wave is strongly converted to L(0,1) and L(0,3) modes, depending also on the curvature parameter of the pipe.

Vibration energy harvesting by magnetostrictive material

A new class of vibration energy harvester based on magnetostrictive material (MsM),Metglas 2605SC, is designed, developed and tested. It contains two submodules: an MsMharvesting device and an energy harvesting circuit. Compared to piezoelectric materials,the Metglas 2605SC offers advantages including higher energy conversion efficiency, longerlife cycles, lack of depolarization and higher flexibility to survive in strong ambientvibrations. To enhance the energy conversion efficiency and alleviate the need of a biasmagnetic field, Metglas ribbons are transversely annealed by a strong magnetic fieldalong their width direction. To analyze the MsM harvesting device a generalizedelectromechanical circuit model is derived from Hamilton’s principle in conjunction withthe normal mode superposition method based on Euler–Bernoulli beam theory. The MsMharvesting device is equivalent to an electromechanical gyrator in series with an inductor.In addition, the proposed model can be readily extended to a more practical case of acantilever beam element with a tip mass. The energy harvesting circuit, whichinterfaces with a wireless sensor and accumulates the harvested energy into anultracapacitor, is designed on a printed circuit board (PCB) with plane dimension25 mm × 35 mm. It mainly consists of a voltage quadrupler, a 3 F ultracapacitor and a smartregulator. The output DC voltage from the PCB can be adjusted within 2.0–5.5 V. Inexperiments, the maximum output power and power density on the resistor can reach200 µW and 900 µW cm−3, respectively, at a low frequency of 58 Hz. For a working prototype undera vibration with resonance frequency of 1.1 kHz and peak acceleration of8.06 m s−2 (0.82 g), the average power and power density during charging the ultracapacitor can achieve576 µW and 606 µW cm−3, respectively, which compete favorably with piezoelectric vibration energy harvesters.

Mass-plate impact parameters for the elastic range

When an impact load is applied laterally to a plate, the deflection gradually spreads over time until the entire plate radius is affected. The determination of effects for a short-duration impact presents substantial mathematical difficulties. The normal-mode superposition is not only awkward but it does not lead to closed-form solutions. Some of the past works gave a possibility of such a solution by assuming a certain flexural wave spreading from the impact point. The complexity involved in that was simplified in this paper. This main part of this work is based on a shear wave approach, by relating the lateral stiffness to the propagation of a shear wave from impact point. This, along with some other simplifications, makes it possible to obtain the peak contact force by using compact expressions. The development is limited to impactor mass not exceeding one-half of the plate mass. The evaluation of rebound velocity or the coefficient of restitution was successfully resolved for only a limited range of parameters. The other subtopic is the quantification of the contact problem itself, which is nonlinear by nature. A transparent method of linearization is proposed, based either on experiment or on Hertzian formulation. The results presented in this paper are compared with the answers from finite-element simulations, physical experiments and with some cases available from literature.

On some possibilities and properties of matched-field geoacoustic inversion in shallow water

The aim of this paper is to estimate possibilities of matched-field geoacoustic inversion (MFI) in shallow water, and to recommend optimal arrangements of signal source and hydrophone array in variety of particular environments. We assumed ocean to be range independent with bottom consisting of homogeneous liquid layers. Sound fields were calculated as superposition of normal modes and continuous spectrum for tonal point source and vertical line array. MFI based on Bartlett processor was used. Possibilities of MFI were characterized by MFI penetration depth, sensitivity to various bottom parameters and non-uniqueness of inverted data. These characteristics were analysed as functions of frequency and the source depth and range for different values of sound attenuation in sediments and for various sound-speed profiles. To estimate possibilities of MFI in real ocean conditions, influence of array tilt and inadequacy of geoacoustic model were analysed. The influence of continuous spectrum was also discussed. Among the major results, optimal source ranges corresponding to the maximum penetration depth of MFI were calculated, and high influence of attenuation in sediments on possibilities of MFI was revealed.

Elastic wave velocities in single-walled carbon nanotubes

This paper reports an atomistic simulation of single-walled carbon nanotubes subjected to harmonic waves, using the molecular structural mechanics method and normal mode superposition. The velocities of longitudinal, transverse, and torsional waves propagating in single walled carbon nanotubes are obtained. The results indicate that the longitudinal wave velocity is roughly twice as much as that of torsional wave. The velocity of the transverse wave in carbon nanotubes is dependent on wave frequency. The nanotube diameter and chirality also have some effects on wave propagation, especially on the torsional wave. The frequency of wave propagating in carbon nanotubes can reach the terahertz range.

Local electric field enhancement near arrays of polarizable cylindrical tubes

We present a multipolar formalism for the local field near an arbitrary array of parallel inhomogeneous cylinders in the long wavelength limit. For very thin tubes the fields may be expressed as a superposition of normal modes. The formalism is applied to arrays of parallel carbon shells filled with silver. A strong enhancement is obtained near the surface plasmon resonance of silver, while that at the resonance of graphite is rather weak. The Raman signal of molecules placed at sites of high field may be enhanced by up to eight orders of magnitude.