Approximate Analytical Relationship Research Articles

Characteristics Analysis of Current-Controlled VSCs for Periodic Forced Oscillations Excited by Grid Frequency Disturbance

Uncontrolled forced oscillations can jeopardize the power system security. Most modeling and analysis methods for forced oscillations in previous publications were linearization methods such that some new nonlinear oscillation phenomenon can not be reproduced powerfully. Based on the considerations above, this paper hopes to provide clearer understandings and recognitions to forced oscillations by theoretically reveal the characteristics. First, a dynamic model of the current-controlled VSCs system based periodic forced oscillations is established to describe the relationships between system parameters and intrinsic quantities. Then, the multi-scale method is introduced to derive the mathematical expressions of approximate analytical solutions and relationship equation. Next, these characteristics, containing jumping phenomenon, frequency shift, doubling frequency response, and frequency coupling, are found and described by analyzing the mathematical expressions. Final, all the results are tested and verified on simulations and experiments.

Skip Re-Entry Trajectory Detection and Guidance for Maneuvering Vehicles.

The re-entry trajectory of maneuvering vehicles with medium to high hypersonic lift-to-drag ratios is generally planned using quasi-equilibrium flight conditions known from Space Shuttles. They may exhibit an oscillation re-entry phenomenon termed skip re-entry when related components or sensors fail. However, conventional re-entry guidance only considers quasi-equilibrium flights and ignores the possibility of the occurrence of an unexpected skip trajectory; this may lead to the failure of the re-entry mission due to a lack of a corresponding guidance strategy. However, the detection of a skip trajectory is the necessary reference for the decision-making of calling a related guidance algorithm that helps improve the safety of vehicle re-entry. Herein, a skip re-entry detection and trajectory control solution is proposed to play an emergency role in the cases of skip re-entry. Firstly, the oscillation frequency characteristics of the linearized re-entry motion equation of a vehicle are analyzed, and an approximate analytical relationship is constructed for skip altitude estimation. Then, the residual deviation between the altitude feedback data and the estimated skip altitude is calculated and compared with the threshold to determine the occurrence of skip re-entry. In addition, a method for controlling the skip re-entry trajectory with the range extension is developed by controlling the bank angle with a fixed angle of attack profile, satisfying the path constraint requirements. The results indicate that the method effectively performs skip re-entry detection and that it can help extend the range of the vehicles in abnormal re-entry scenarios, keeping the flight within the path constraints and guiding it to the expected location.

Can the Earth’s harmonic spectrum be derived directly from the stochastic inversion of global travel-time data?

<p>A set of seismic observations which all sample the same structure in the same way should have zero variance. This is naturally the case if all sources are in the same place, and the data are recorded by the same station. If sources and/or receivers are not in the same place, but close to one another, variance will generally be nonzero, but small. Variance might become large if the sampled region of the Earth contains heterogeneities whose spatial wavelength is comparable to the distances between sources and between receivers (and thus between the corresponding ray paths). The travel-time variance of a “bundle” of seismic rays thus reflects the degree of complexity of the sampled region of the medium. We apply this simple principle to real seismic databases, attempting to constrain the spherical harmonic spectrum of Earth’s structure without having to derive a tomographic model. This results in a reduction of the dimensionality of the solution space, and hence of computational costs. This approach allows to constrain the statistical properties, rather than exact geographic locations of structural features; knowing the statistics of Earth’s structure is most valuable for many fundamental geodynamic questions. We follow an earlier study by Gudmundsson et al. [1990] to find an approximate analytical relationship between averaged variance and harmonic spectrum; this allows us to determine the latter from a measurement of the former via a linear least squares inversion. Our analysis shows that the variance of ray bundles associated with large geographic extent of source/receiver bins is sensitive to low-degree spectral power, and vice-versa for small bins/high harmonic degrees. The method is accordingly ineffective at very low harmonic degrees, associated with an inherently limited number of source-receiver bins. We conduct a suite of inversions of both real and synthetic seismic data sets to evaluate the resolving power of our algorithm, and attempt to identify a range of harmonic degrees where the method is robust. Our results indicate that the resolution of the Earth’s spectrum afforded by the method presented here is inferior to that of classical tomography.</p>

An analytical determination of the viscosity of liquids

I show that one can obtain an exact analytical expression for the viscosity of a liquid by solving the equation of motion for a spherical ball falling through it. I also deduce an approximate analytical relationship suitable for an experimental determination of the viscosity of liquids.

The nonlinear evolution and approximate scaling of directionally spread wave groups on deep water

The evolution of steep waves in the open ocean is nonlinear. In narrow-banded but directionally spread seas, this nonlinearity does not produce significant extra elevation but does lead to a large change in the shape of the wave group, causing, relative to linear evolution, contraction in the mean wave direction and lateral expansion. We use the nonlinear Schrödinger equation (NLSE) to derive an approximate analytical relationship for these changes in group shape. This shows excellent agreement with the numerical results both for the NLSE and for the full water wave equations. We also consider the application of scaling laws from the NLSE in terms of wave steepness and bandwidth to solutions of the full water wave equations. We investigate these numerically. While some aspects of water wave evolution do not scale, the major changes that a wave group undergoes as it evolves scale very well.

Determination of the wave mode contribution into the ULF pulsations from combined radar and magnetometer data: Method of apparent impedance

To identify the physical nature of Pc5 pulsations and to determine relative contributions of different MHD modes into their structure, we introduce the method of apparent impedance. An approximate analytical relationship from the theory of ULF wave transmission through a thin ionosphere can be compared with the measured ratio between simultaneous ionospheric electric and ground magnetic fields. The analytical expressions show that the impedances of Alfven and compressional modes should be essentially different. To validate this technique, we have analyzed global Pc5 pulsations during the recovery phase of the strong magnetic storm on October 31, 2003 using ground magnetometricdata and tri-static EISCAT radar. This radar facility gives a possibility to calculate vectors of the F-region ionospheric plasma flow and electric field. From these observations we conclude that the global Pc5 pulsations during storm conditions above the ionosphere are predominantly composed from Alfven waves with a small contribution of fast compressional mode.

Relating the maximum radial stress on pile shaft to pile base resistance

An approximate analytic relationship is developed between the maximum radial stress on the shaft of a displacement pile in sand and the base resistance of the pile. Using the cavity expansion analogy, together with a confined failure mechanism, the ratio between the two quantities, defined as a factor St, is established as a function of the friction angle, shear stiffness, compressibility and mean effective stress of the sand near the pile tip. It is shown that, given otherwise identical input parameters, the value of St will decrease with increasing friction angle, and with decreasing mean stress level. It also tends to decrease with an increase in relative density. It is predicted that St has typical values between 0·03 and 0·05, in broad agreement with the range of empirically derived values in the literature. The relationship also predicts that St may take much higher values (∼0·1) for piles installed in dense sand or in highly compressible sand. Because of the analytical nature, the established relationship provides useful insights into the mechanisms involved and important implications for design practice.

Error cancellation in the semiclassical calculation of the scattering length

We investigate the effects of two approximations concerning long range dispersion forces that are made in the derivation of the semiclassical formula for the scattering length of a pair of neutral atoms. We demonstrate numerically, using a published model interaction potential for a pair of Cs atoms in the \(^3{\rm\Sigma}_{\rm u}^+\) molecular state, that the subsequent long range errors tend to cancel and we show, from an approximate analytical relationship, that the first order errors do indeed largely cancel. We suggest a hybrid method that combines quantum mechanical and semiclassical calculations. We explore its use in finding the scattering lengths of 7Li atoms and 133Cs atoms interacting via the X\(^1{\rm\Sigma}^+\) and a\(^3{\rm\Sigma}^+\) molecular potentials and we use it to demonstrate that the semiclassical formula fails for cold collisions of H atoms in the X\(^1{\rm\Sigma}_{\rm g}^+\) molecular state because of the long range errors rather than because of inadequacies in describing the motion over the potential well semiclassically.

Frequency coupling in bistable system and the mechanism of stochastic resonance

We study the motion of a particle in a bistable potential in the presence of single frequency signal and dual-frequency signal, and give the approximate analytical relationship between response peak-to-peak value and driven peak-to-peak value of the bistable system under the control of dual-frequency signal. The paper also reveals a peculiar phenomenon of the non-linear system, that is frequency coupling of different frequencies and the energy infiltration from the frequency of input to the other. The analysis is carried out from the view point of dynamics mechanism and the frequency spectrum distribution. It deepens the understanding of the mechanism of stochastic resonance in the bistable system. The results of numerical simulation prove the validity of theoretical analysis.

Asymptotically Self-Similar Propagation of Spherical Ionization Waves

It is shown that a new type of self-similar spherical ionization waves may exist in gases. All the spatial scales and the propagation velocity of such waves increase exponentially with time. The conditions for existence of these waves are established, their structure is described, and approximate analytical relationships between the principal parameters are obtained. It is notable that spherical ionization waves can serve as the simplest, but structurally complete and physically transparent model of a streamer in a homogeneous electric field.

Dual‐porosity modelling of the pore structure and transport properties of a contaminated soil

SummaryWe have developed a new method to characterize the pore structure of mineral soils. We combined data from the analysis of back‐scattered scanning electron microscope (BSEM) images of resin‐impregnated pore‐casts, and mercury intrusion porosimetry (MIP) data, with analytical percolation models and inverse modeling algorithms. The pore space is regarded as a dual‐pore network consisting of a primary Euclidean pore‐and‐throat network and a secondary, fractal, pore system that is accessed through primary pores. The digitized 2‐D BSEM images of resin‐impregnated soil samples are employed to determine the autocorrelation function. The Fourier transform of this function provides the small‐angle neutron scattering (SANS) intensity function, which is extended by using the surface fractal dimension obtained from high‐pressure MIP data. Inversion of the extended scattering intensity function produces the volume‐based radius distribution function of spherical pore bodies (PBRD). The complete volume‐based PBRD is fitted with a composite number‐based PBRD composed of a lognormal primary PBRD and a power (fractal) secondary PBRD with upper and lower cut‐offs. Based on the concepts of invasion percolation, an analytic mathematical model that describes Hg intrusion into dual pore networks is developed. The complete PBRD and pore‐throat radius distribution (PTRD) functions of the primary network along with the drainage accessibility functions (DAFs) of the primary and secondary pore networks are estimated with inverse modelling of the Hg intrusion curve. Based on critical path analysis of percolation theory, approximate analytical relationships are developed to calculate explicitly the absolute permeability and electrical formation factor from the geometrical and topological parameters of the primary pore network. The method is demonstrated with application to four soil samples.

An approximate method of determination of maximum strain hardening levels in metals under nonproportional low-cycle loading

Using the findings of analysis of deformation curves for metallic materials under static and cyclic loading, an approximate method is put forward for the determination of maximum strain hardening levels in the case of non-proportional low-cycle loading with strain monitoring. Based on the correlation between strain hardening data obtained from the static and proportional and nonproportional cyclic deformation curves, an approximate analytical relationship is built up which allows for predicting maximum strain hardening levels under nonproportional low-cycle loading.

Electrical contact resistance theory for conductive rough surfaces separated by a thin insulating film

An electrical contact resistance (ECR) theory is presented for conductive and rough (fractal) surfaces separated by a thin insulating film, which is treated as an energy barrier that impedes current flow due to the electric-tunnel effect. The analysis yields insight into the effects of film properties, current flow, surface topography, mechanical properties, and contact load on the ECR. It is shown that the variation of ECR with the contact load is less pronounced than that observed in the absence of an insulating layer due to the intrinsic voltage dependence of the tunnel resistance and the associated voltage compensation mechanism. The effect of nonohmic behavior on the relationships of the ECR with the contact load and the real contact area is discussed and results are compared with approximate analytical relationships developed herein. The relationship between the real contact area and the ECR depends on the current intensity and film properties and is independent of the surface topography and mechanical properties. Approaches for determining the surface roughness, mechanical properties, insulating film properties, and real contact area from ECR measurements are interpreted in light of the theory developed.

A model for instability growth in accelerated solid metals

We present an approximate analytical dispersion relationship for elastic–plastic acceleration-driven instability growth. In this model, the accelerated solid behaves like a viscoplastic after its elastic yield strength is exceeded, with the viscosity inversely proportional to the strain rate. We have applied this model, or a 1993 model of shock-driven viscous instability growth, where applicable, to perturbation growth measurements made in three separate types of experiments: High-explosive (HE)-driven planar Al plates, HE-driven implosions of steel cylinders, and planar Al foils driven indirectly by Lawrence Livermore National Laboratory’s Nova laser. We have also compared the analytical modeling of these experiments with simulations done with a two-dimensional Lagrangian radiation-hydrodynamics computer code containing an elastic–plastic constitutive model. We find that for the moderate strain rates of the HE experiments, the simulations and analytical modeling of perturbation growth agree with each other and with the data, using an equivalent plastic viscosity consistent with the von Mises plasticity criterion. For the high strain rates of the Nova experiments, on the other hand, the early-time growth data is consistent with viscoplastic growth, with viscosity ten to a hundred times less than the von Mises plastic viscosity for nominal strength. This observed initial material weakening is followed by a transition to a strengthened state to match the late-time growth data, which we show to be consistent with a “relaxation” hypothesis in which plastic flow at high strain rate is confined to discrete shear bands. We also show under what conditions the perturbation growth is independent of initial amplitude.

Simulating Non-Isothermal Water Vapour Transfer: An Experimental Validation on Multi-Layered Building Components

The aim of this study is to validate different analytical relations used in hygrothermal simulations for the material properties. Therefore, a valida tion experiment on four types of flat roofs has been set up at the laboratory. All rele vant material properties of the individual material layers were determined before hand. During the experiment, temperatures, heat fluxes and moisture contents were measured. The measured data were then compared with numerical simulation results. By repeating the simulations with different expressions for the vapour per meability and for the sorption isotherm, the influence of approximate analytic rela tionships was investigated.

Linear evolution of baroclinic waves in saturated air

A primitive-equation model which includes a moisture parametrization that assumes that ascending air is always saturated, has been used to study the characteristics of moist baroclinic waves, exploring their dependence on the Richardson number, besides zonal and meridional wavelength. Growth rates are shown to increase, in consequence of latent-heat release, in a larger measure for the meridionally structured waves than for the two-dimensional ones. Approximate analytic relationships for the normal-modes growth rates, based on the concept of a weighted average of dry and saturated parameters, compare favourably with the numerical results. Asymmetric horizontal structures, previously obtained in quasi-geostrophic approximation, are shown to be consistent with potential-vorticity generation by diabatic heating in the updraught.

Linear evolution of baroclinic waves in saturated air

AbstractA primitive‐equation model which includes a moisture parametrization that assumes that ascending air is always saturated, has been used to study the characteristics of moist baroclinic waves, exploring their dependence on the Richardson number, besides zonal and meridional wavelength. Growth rates are shown to increase, in consequence of latent‐heat release, in a larger measure for the meridionally structured waves than for the two‐dimensional ones. Approximate analytic relationships for the normal‐modes growth rates, based on the concept of a weighted average of dry and saturated parameters, compare favourably with the numerical results. Asymmetric horizontal structures, previously obtained in quasi‐geostrophic approximation, are shown to be consistent with potential‐vorticity generation by diabatic heating in the updraught.

Resistivity index of fractional wettability porous media

The effects of fractional wettability on electrical resistivity index curves of porous media are investigated using pore network models. A bond percolation-and-fractal roughness model is used to simulate the oil/water drainage of the conventional porous plate method in pore networks composed of randomly distributed `strongly water-wet' and `strongly oil-wet' capillaries. Based on universal scaling laws of percolation quantities, effective medium approximation and fractal geometry, approximate analytic relationships are developed with respect to the dependence of the resistivity index, capillary pressure and saturation exponent on certain microstructural properties of the pore space and surface fractional wettability over the various water saturation regions. The simulated data are fitted to two-exponent power laws, which in turn are evaluated as macroscopic conceptual models of the resistivity index. At high water saturations, the saturation exponent becomes a strongly increasing function of the fraction of oil-wet pores when the value of this parameter exceeds the percolation threshold of the lattice network and oil percolates spontaneously through network joining clusters of oil-wet pores. At intermediate water saturations, the saturation exponent is a moderately increasing function of the fraction of oil-wet pores, whereas the slope of the capillary pressure curve remains almost unaltered to variations of wettability. At low water saturations, as the fraction of oil-wet pores becomes quite large, permanent trapping of water may occur with result that both the saturation exponent and the slope of the capillary pressure curve tend to infinity at the limit of irreducible water saturation. The exponents of the phenomenological models of the resistivity index change significantly with fractional wettability and are consistent with the values of the saturation exponent obtained with the approximate analytic relationships.

Grapho-analytic methods for the parametric estimation of the characteristics of bending-torsion flutter of airfoils

We study the possibility of estimation of the critical velocity and frequency of bending-torsion flutter of an airfoil by using grapho-analytic methods and obtain approximate analytic relationships for the parameters of the airfoil and air flow, which enable us to represent these parameters in a graphic form convenient for subsequent analysis.

Fatique and damage-tolerance analysis of composite laminates: Stiffness loss, damage-modelling, and life prediction

The prediction of fatigue life and evaluation of progressive damage for general composite laminates are studied analytically. From theories of damage tolerance, residual-modulus degradation, and residual-strength d degradation, a simple and approximate approach is proposed for the prediction of progressive stiffness loss, matrix-crack density, and delamination area in terms of tension-tension fatigue load and number of cycles for general laminates containing 0° plies. The proposed approach provides four choices for predicting tension-tension allowable fatigue life and for assessing fail-safety for structures made of composite laminates, namely, a residual-modulus criterion, a matrix-cracking criterion, a delamination-size criterion, and a residual-strength criterion. The approximate analytical relationship between S-N curves for general laminates containing 0° plies and for a unidirectional 0°-ply laminate is proposed. Three glass/epoxy laminates of [0/90] s, [±45/0/90] s , and [0/±45] s lay-ups, and one [0/90±45] s T300/5208 graphite/epoxy laminate under tension-tension fatigue are used to illustrate and verify the proposed approach. The analytical results are in good agreement with experimental data.