Frequency-domain Formulation Research Articles

Boundary element frequency domain formulation for dynamic analysis of Mindlin plates

AbstractA new boundary element formulation for analysis of shear deformable plates subjected to dynamic loading is presented. Fundamental solutions for the Mindlin plate theory are derived in the Laplace transform domain. The characteristics of the three flextural waves are studied in the time domain. It is shown that the new fundamental solutions exhibit the same strong singularity as in the static case.Two numerical examples are presented to demonstrate the accuracy of the boundary element method and comparisons are made with the finite element method. Copyright © 2006 John Wiley & Sons, Ltd.

Signal-to-noise ratio and frequency analysis of continuous loop averaging deconvolution (CLAD) of overlapping evoked potentials

In this study, a frequency domain formulation of continuous loop averaging deconvolution (CLAD) of overlapping evoked potentials is developed and applied for the extraction of transient responses from recordings obtained at high stimulation rates. This formulation allows for a faster execution of CLAD by using fast Fourier transform algorithms. The frequency characteristics of the deconvolution filter depends exclusively on the stimulus sequence and determines whether the noncoherent noise is amplified or attenuated in different frequencies. A formula for calculating the signal-to-noise ratio (SNR) achieved by the deconvolution process is developed. The newly developed theory and the methodology is applied to the extraction of the auditory brainstem and middle latency responses using various sequences. The effects of the sequence used and the number of sweeps averaged in ongoing acquisition on SNR are examined by using single sweep recordings. The results verify the deconvolution theory and the methodology and show its limitations. Depending on the frequency characteristics of the sequence, the deconvolution process can amplify or attenuate the EEG noise. Proper selection of the stimulus sequence can increase the SNR enhancement obtained with conventional averaging.

A discrete-time uniform geometrical theory of diffraction for the fast transient analysis of scattering from curved wedges

In this paper a discrete time (DT) representation of Maxwell's equations is employed to describe the time domain (TD) Maxwell's equations in terms of matrix equations analogous to time harmonic Maxwell's equations, which allows one to conveniently develop many TD solutions based on their frequency domain (FD) formulations. It was then employed to develop a TD version of the uniform geometrical theory of diffraction (UTD) (referred as DT-UTD) for the efficient electromagnetic transient analysis of scattering from perfectly conducting curved wedges. This DT-UTD retains the advantages of its corresponding FD UTD solution in several aspects including the form, ray physical picture and definitions of ray parameters. Furthermore, the transformation of the FD-UTD to the DT-UTD formulation can be easily achieved via a simple interpretation of notation. Numerical examples are presented to validate and illustrate the utilizations of this DT-UTD.

Alternate methods for characterizing spectral energy inputs based only on driving point mobilities or impedances

This article analyses three characterization methods for spectral energy inputs to a linear time-invariant system. Analytical frequency domain formulations are examined for discrete vibratory systems and one-dimensional continuous structure (undergoing longitudinal or flexural motions) given a harmonic force excitation. Two existing methods that have been proposed by prior researchers are first critically examined. In particular, the driving point transfer functions and their derivatives with respect to frequency are analyzed for an appropriate application to the energy characterization scheme and to determine the sources of error. Then, a new (third) scheme is proposed that is more suitable over low and mid frequency regimes, based on a proper interpretation of the driving point mobilities or impedances and their derivatives. The new method is found to be insensitive to the driving point mobility or impedance formulations, unlike the existing methods. It does yield consistent results, without requiring a prior knowledge of the transfer functions. Finally, the role of structural loss factor has been clarified in the context of the stated problem.

Fast numerical analysis of dielectric resonators with perturbed rotational symmetry

A fast finite difference frequency domain (FDFD) formulation for obtaining resonant frequencies of structures with perturbed rotational symmetry is described. The formulation combines a three-dimensional (3-D) eigenfunction function expansion algorithm with 3-D FDFD in the cylindrical coordinate system. Numerical tests have showed that the technique gives high accuracy results and significant CPU time reduction.

Kinetic and spectral descriptions of polarized atomic radiative emission from plasmas

A reduced-density-matrix description has been developed for the investigation of polarized radiative emission during single-photon transitions from bound and autoionizing states of ionized atomic systems in the presence of a general arrangement of static (or quasi-static) electric and magnetic fields. Particular emphasis has been given to excitation of the atomic states by electrons with an anisotropic velocity distribution, which can be produced in an electron–ion beam experiment or in a non-equilibrium plasma environment. It is desirable to consider the coherent excitation of a particular subspace of the atomic bound or autoionizing states, which can occur as a result of sufficiently intense electromagnetic interactions. A general expression for the matrix elements of the detected-photon density operator provides a unified framework for the analysis of the spectral intensity, angular distribution, and polarization of the Stark–Zeeman patterns. From a unified development of time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations of the reduced-density-matrix approach, the non-equilibrium atomic-state kinetics and the homogeneous spectral-line shapes are self-consistently described.

Reference Current Computation Methods for Active Power Filters: Accuracy Assessment in the Frequency Domain

This paper focuses on the steady-state response of existing methods for computing the reference current of active power filters. For each class of methods, the main source of discrepancy between the load harmonic current and the computed reference current is identified and the frequency spectrum of the resulting error is analytically determined. Although this topic has been partially addressed in previous publications, the proposed frequency-domain approach provides valuable qualitative information about how the errors are produced and distributed, which is masked when the analysis is carried out in the time domain. First, the frequency-domain formulation is separately presented for each method. Then, a comparison of the resulting errors is performed on a case study. Finally, some experimental results are given to validate the proposed frequency-domain analysis.

Frequency domain formulation of active parametric deformable models

Active deformable models are simple tools, very popular in computer vision and computer graphics, for solving ill-posed problems or mimic real physical systems. The classical formulation is given in the spatial domain, the motor of the procedure is a second-order linear system, and rigidity and elasticity are the basic parameters for its characterization. This paper proposes a novel formulation based on a frequency-domain analysis: The internal energy functional and the Lagrange minimization are performed entirely in the frequency domain, which leads to a simple formulation and design. The frequency-based implementation offers important computational savings in comparison to the original one, a feature that is improved by the efficient hardware and software computation of the FFT algorithm. This new formulation focuses on the stiffness spectrum, allowing the possibility of constructing deformable models apart from the elasticity and rigidity-based original formulation. Simulation examples validate the theoretical results.

Reduced-density-matrix descriptions for coherent electromagnetic interactions in quantized many-electron systems

A reduced-density-matrix description is developed for coherent linear and non-linear electromagnetic interactions of quantized electronic systems in the presence of environmental decoherence and relaxation phenomena. Applications of interest include many-electron atomic systems and semiconductor materials. Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations of the reduced-density-matrix approach are developed in a unified manner. The general non-Markovian formulations provide a rigorous foundation for the systematic introduction of the commonly used Born (lowest-order perturbation-theory) and Markov (short-memory-time) approximations. A preliminary semiclassical perturbation-theory treatment of the electromagnetic interaction is introduced, based on a coupled hierarchy of relationships for the field components of the reduced density operator.

Global control of a vibrating plate using a feedback-controlled inertial actuator

Strategies for the suppression of plate vibration are investigated by considering approximations to the equivalent impedance of power-minimizing vibration controllers. The total power transmitted to a plate by both a primary and a secondary point force is minimized and the equivalent impedance presented by the secondary source to the plate is considered. A novel device for active vibration control, based on an inertial actuator with displacement sensor and local PID controller and an outer velocity feedback control loop, is used to control the vibrating flexible plate. The impedance presented to the plate by this actuator is compared with the equivalent impedance of the optimal active control system. A frequency-domain formulation is used to analyse the stability and performance of an active vibration suppression system using this modified inertial actuator. The results of an experimental study of active vibration suppression on a flexible plate using the modified inertial actuator are then described. Theory and experiments agree well, demonstrating the effectiveness of the modified inertial actuator.

Active vibration isolation using an inertial actuator with local displacement feedback control

The design of inertial actuators with local displacement feedback control and their use in active vibration isolation systems is considered. Unlike reactive actuators, inertial actuators do not need to react off a base structure and can therefore be directly installed on a vibrating structure. However in order to guarantee good stability margins in the active isolation controller, the actuator resonance must have a low natural frequency and it must be well damped. However, the need to have an inertial actuator with a low resonance frequency leads to unwanted static deflections of the actuator proof-mass. The use of integral displacement feedback as a local loop within the actuator provides self-levelling capabilities for the inertial actuator proof-mass, thus overcoming the static deflection problem. A novel device for active vibration control, based on an inertial actuator with displacement sensor and local PID controller, is described and its performance is demonstrated experimentally. It is found that the natural frequency and damping of the actuator can also be changed substantially with such a controller, thus allowing an inertial actuator to be customised for a specific application. A frequency-domain formulation is then used to analyse the stability and performance of an active isolation system using the modified inertial actuator and an outer velocity feedback control loop. The plant response, from force actuator input to sensor output, is derived in terms of the mechanical mobilities of the equipment structure being isolated and the vibrating base structure, and the mechanical impedance of the intervening mount. The results of an experimental study of active vibration isolation using a modified inertial actuator are then described. Theory and experiments agree well, demonstrating the effectiveness of the modified inertial actuator.

Boussinesq evolution equations: numerical efficiency, breaking and amplitude dispersion

This paper deals with the possibility of using methods and ideas from time domain Boussinesq formulations in the corresponding frequency domain formulations. We term such frequency domain models “evolution equations”. First, we demonstrate that the numerical efficiency of the deterministic Boussinesq evolution equations of Madsen and Sørensen [Madsen, P.A., Sørensen, O.R., 1993. Bound waves and triad interactions in shallow water. Ocean Eng. 20 359–388] can be improved by using Fast Fourier Transforms to evaluate the nonlinear terms. For a practical example of irregular waves propagating over a submerged bar, it is demonstrated that evolution equations utilising FFT can be solved around 100 times faster than the corresponding time domain model. Use of FFT provides an efficient bridge between the frequency domain and the time domain. We utilise this by adapting the surface roller model for wave breaking to frequency domain evolution equations. An equation for the variation of the mean water level is derived. Results for regular and irregular waves are presented and compared to results of conventional breaking formulations for evolution equations as well as for results of the corresponding time domain model. Emphasis is given to the shape of the breaking waves. The amplitude dispersion of evolution equations is analysed using a third-order perturbation approach. It is found to exceed the amplitude dispersion of the corresponding time domain model, and the approximation causing this deviation is pinpointed.

Adaptive Spectral Strain Estimators for Elastography

In conventional elastography, internal tissue deformations, induced by external compression applied to the tissue surface, are estimated by cross-correlation analysis of echo signals obtained before and after compression. Conventionally, strains are estimated by computing the gradient of estimated displacement. However, gradient-based algorithms are highly susceptible to noise and decorrelation, which could limit their utility. We previously developed strain estimators based on a frequency-domain (spectral) formulation that were shown to be more robust but less precise compared to conventional strain estimators, In this paper, we introduce a novel spectral strain estimator that estimates local strain by maximizing the correlation between the spectra of pre- and postcompression echo signals using iterative frequency-scaling of the latter; we also discuss a variation of this algorithm that may be computationally more efficient but less precise. The adaptive spectral strain estimator combines the advantages of time- and frequency-domain methods and has outperformed conventional estimators in experiments and 2-D finite-element simulations.

Spectral Element Method for LInear Fan Tone Noise Radiation

A numerical method for prediction of acoustic spinning mode radiation from turbofan inlets is presented. Sound propagation is modeled by the linearized mass conservation equation for irrotational flows and solved in the frequency domain. The mean flow through the inlet is obtained as a solution of the full potential equation. Both the mean-flow and the acoustic problem are approximated by Galerkin projection in spectral element spaces of continuous piecewise polynomials defined on the same grid. The Gauss-Chebyshev-Lobatto points within the elements are generated via transfinite interpolation and CAD projection procedures embedded within the code. The linear algebraic systems obtained are then solved using either direct or sparse iterative solvers based on the message passing interface standard for interprocessor communication. The singularity appearing in the acoustic integrals on the symmetry axis is treated by the use of a collocation operator based on the Gauss-Chebyshev, instead of the Gauss-Chebyshev-Lobatto, points. To eliminate reflections from the radiation boundaries, a novel frequency-domain formulation of the matched-layer technique, wherein waves entering the layer are exponentially damped, is proposed. The overall computing procedure is first validated on a tone radiation problem from a semi-infinite cylinder and then applied to an experimental JT15D turbofan inlet setup.

Macromodels in the frequency domain analysis of microwave resonators

We present a new technique of incorporating macromodels into the frequency domain formulation of grid based methods. Unlike previous methods, the new scheme does not introduce any frequency dependent elements into the system matrix and thus can be used in the analysis of resonator problems. Numerical results show an increase in accuracy of computations in a wide frequency range while convergence properties of the underlying matrix eigenvalue solver are not affected.

A comparison of frequency-domain transfer function model estimator formulations for structural dynamics modelling

The purpose of this paper is to present the results of a comparative study of various formulations of a frequency-domain least-squares (LS) estimator. This study was the basis for the optimization of the LS algorithm for modal parameter estimation. Nowadays, modal analysis studies the dynamical behaviour of very complex structures, which requires both robustness for high model orders and efficient numerical properties to analyse extensive data sets that are obtained during high channel-count modal testing. This paper focuses on the optimal choice between various transfer function models and a weighted formulation of the LS problem by taking also a measure for the noise on the data into account. The evaluation is based on a practical case study, discussing the dynamical modelling of slat tracks of an Airbus A320 commercial aircraft. This safety critical component is characterized by a complex modal behaviour requiring both high model orders and a high spatial resolution in order to achieve an accurate model.

Active vibration isolation using an inertial actuator with local force feedback control

The design of inertial actuators with local force feedback control and their use in active vibration isolation systems is considered. Unlike reactive actuators, inertial actuators do not need to react off a base structure and can therefore be directly installed on a vibrating structure. In order to guarantee good stability margins in the active isolation controller, however, the actuator resonance must have a low natural frequency and it must be well damped. The behaviour of an inertial actuator with different local force feedback control schemes is first analysed, and it is shown that a phase-lag controller has a good stability margin and can effectively damp the actuator resonance using relatively low gains, compared with a direct force feedback or integrated force feedback controller. A frequency-domain formulation is then used to analyse the stability and performance of an active isolation system using an inertial actuator with local force feedback control and an outer velocity feedback control loop. The plant response, from force actuator input to sensor output, is derived in terms of the mechanical mobilities of the equipment structure being isolated and the vibrating base structure, and the mechanical impedance of the intervening mount. An experimental study of active vibration isolation using an inertial actuator with local feedback control was then carried out. Theory and experiments agree well, demonstrating the effectiveness of the phase-lag controller. However, the need to have an inertial actuator with a low resonance frequency leads to problems with static deflections.

Mobility analysis of active isolation systems

A frequency-domain formulation is used to analyze the stability and performance of an active vibration isolation system which uses feedback control. The active mount is modelled as a single-axis force actuator in parallel with a passive spring and damper. The feedback sensor measures either the absolute velocity of the equipment to be isolated at one end of the mount, or the integral of the transmitted force through the mount. The plant response, from force actuator input to sensor output, is derived for these two cases in terms of the mechanical mobilities of the two structures connected by the active mount. The limits of the phase of the plant response are derived for the two feedback strategies and these are used to explain the stability and performance of several specific examples of active isolation systems. It is shown that, in the absence of actuator and sensor dynamics, the integrated force feedback system is unconditionally stable. The stability of the absolute velocity feedback system is, however, threatened if the vibrating base structure becomes very mobile, with a small effective mass, at the same frequency as the equipment structure becomes very stiff. By quantifying the conditions under which velocity feedback systems can become unstable, these conditions can be avoided. If the stability of an absolute velocity feedback system can be assured, it is shown to be more effective at controlling resonances caused by equipment dynamics than integrated force feedback.

Initial conditions in frequency-domain analysis: the FEM applied to the scalar wave equation

The present paper describes a procedure to consider initial conditions in the frequency-domain analysis of continuous media discretized by the FEM. The frequency-domain formulation presented here is based on a standard DFT procedure, the FFT algorithm being employed to transform from time to the frequency domain and vice versa. The standard Galerkin finite element method (displacement model) is used to replace the original differential governing equation by an integral equation amenable to numerical solution. The scalar wave equation (one- and two-dimensional) is used to illustrate the proposed approach. At the end of the paper, examples of wave propagation for a taut string, a one-dimensional rod and a membrane are presented to illustrate the robustness of the formulation presented here.

Short‐pulse radiation by a sequentially excited semi‐infinite periodic planar array of dipoles

This paper deals with the fourth in a sequence of canonical problems aimed toward an understanding of the time domain (TD) behavior of wideband‐excited sequentially pulsed planar periodic finite arrays of dipoles, which play an important role in a variety of practical applications. The present investigation of sequentially pulsed semi‐infinite planar dipole arrays extends our previous studies of sequentially pulsed infinite and semi‐infinite line dipole arrays and of infinite planar dipole arrays. The discrete element‐by‐element radiations are converted collectively to radiations from a series of Floquet wave (FW)‐modulated truncated smooth equivalent aperture distributions, and to corresponding FW‐modulated edge diffraction. After a summary of necessary results from the earlier studies, emphasis is placed on the new truncation‐induced TD results and interpretations, which are extracted via phenomenology‐matched high‐frequency asymptotics from rigorous frequency and time domain formulations parameterized in terms of the dispersive FW instantaneous frequencies and wave numbers. As in our previous studies, the outcome is a numerically efficient, physically incisive algorithm whose accuracy is verified preliminarily by application to a pulsed planar strip array of dipoles.