Small Perturbation Assumption Research Articles

Error on the constitutive relation in the nonlinear formulation of elastic beams: the plane case

The error measure we propose in this study is based on the mechanics concept of error on the constitutive relation. The error on the constitutive relation has already been used for the updating of elastic structures in dynamics, under the small perturbation assumption. Here, we extend this concept to the context of geometrically nonlinear elastic beams (large rotations and finite strains). It is shown that the proposed strategy, which uses measured experimental data, enables us to locate damaged substructures or, if the quality of the model is to be checked, the method provides the location of the zones of modelling errors. The formulation and its derivation are developed and a complete algorithmic treatment is presented. The implications of the formulation of the errors location discussed in this paper are illustrated through a set of numerical examples.

Identification of non-linear derivative models from Bo 105 flight test data

AbstractThis paper describes the results of a study focusing on the possibility of identifying nonlinear helicopter models in the time domain. In recent years, identification techniques working mainly in the frequency domain were applied to estimate the parameters in helicopter models. Recently, the time domain identification method of the DLR was improved to allow the identification of fully nonlinear models. To investigate the applicability of this method to helicopters, a nonlinear derivative model with explicit equations for the individual blade flapping angles was formulated. In comparison to the linear derivative models nonlinear models are physically more realistic and are not restricted by small perturbation assumptions. Although a large number of unknowns had to be determined, a nonlinear model was successfully identified.

Towards a nonperturbation transport theory in hetereogeneous aquifers

A large body of existing theories of flow and contaminant transport in aquifers ignore the presence of recharge, eliminate the boundary conditions, neglect transient conditions in groundwater flow, conceive hydraulic gradients as linear, and require parameter variability to be stationary and Gaussian. The most outstanding and difficult to justify assumption is the subjective “small” size of the stochastic terms (i.e., small perturbation methods), which usually is forced by considering the logarithm of the hydraulic conductivity. Several problems in flow and contaminant subsurface hydrology, such as the enhanced dispersion parameters with plume size or time after injection, remain to be observed in the light of a stochastic theory that allows a more realistic consideration of physical and hydrologic properties. In this article, an attempt is made to reformulate a contaminant transport equation (the variable dispersion equation, VDE) with transport parameters in terms of regional hydrologic and aquifer hydraulic properties, such as recharge rate, spatially random transmissivity, hydraulic gradient, aquifer thickness, and soil porosity. Subsequently, a general analytic procedure, the method of decomposition, is used to derive a solution to the VDE. This procedure does not require small perturbation, logarithmic transformations, or specific probability law assumptions. Comparison tests with existing theoretical and field results are given. The tests illustrate the enhanced dispersion and shifting concentration effects produced by the variable dispersion equation. Finally a generalization of the method to nonstationary dispersion in three-dimensional domains is proposed.

Ramped oscillatory testing: A new technique to examine dynamic strain ageing

A new experimental technique for examining dynamic strain ageing in solid solutions is proposed. It is based on the imposition of an oscillatory perturbation on a tensile specimen monotonically deforming under strain or stress control. Experimental measurements on an AlMgSi alloy show that the attendant oscillatory response has characteristic features which may be used to detect and characterize dynamic strain ageing. A constitutive model which accounts for dynamic strain ageing is used to provide a theoretical basis for the analysis. A closed-form solution of the governing equations is obtained under the assumption of small oscillatory perturbations. A full numerical solution is given for the AlMgSi alloy which was tested using the technique proposed. Good agreement of the predictions of the model with the experimental results is obtained.

Recent findings on Kelvin's equilibria

The linearized stability analysis is applied to investigate the wave behavior in a water vortex produced in a cylindrical tank with a flat disk rotating at the bottom. Two flow cases are considered herein. The first case deals with waves developed on the free surface of a hollow vortex, while the second with waves generated in the core of a Rankine vortex. It is evident from the analysis that the experimental dispersion velocity approaches the calculated one when the wave amplitude becomes smaller. The latter is consistent with the small perturbation assumption that is inherent in the theory. For the case where the core is flooded, the presence of a cylindrical wall is shown to enhance the wave speed. A hypothesis as to how the core develops in the mixed state regions is proposed. The graphical simulations appear to predict reasonably well the main features of the observations.

An error study on some linear reconstruction algorithms for electrical impedance tomography.

An error analysis on the Poisson-type linear reconstruction algorithms is performed. Three types of error are discussed and it is found that re-scaling error is dominant in most cases. It is also found that the commonly used small perturbation assumption in obtaining the Poisson's equation often fails but, due to the dominance of the re-scaling error, the reconstruction of the relative impedance distribution is still feasible by using the linear algorithms. The error analysis leads to a useful understanding of the mechanism of linear reconstruction approaches.

Sensitivity of time domain reflectometry measurements to lateral variations in soil water content

The spatial responses of two types of time domain reflectometry (TDR) probes used for measuring the water content of soils and other porous materials are examined. We investigate the effect of spatial variation in the water content of the sample on the measured value of the water content for both the coaxial cylinders probe and the two parallel wires probe. The response of the instrument to water content distributions that are perturbed slightly from a uniform distribution is calculated, using the conventional electrostatic treatment of electromagnetic wave propagation in the transverse electro‐magnetic (TEM) mode. Under this small perturbation assumption, we show that the spatial weighting function is approximately proportional to the distribution of electromagnetic energy between the electrodes for uniform water content, calculated from a solution of Laplace's equation for each probe type. For coaxial probes, most of the energy (and hence most of the measurement sensitivity) is concentrated around the inner cylinder in a “skin effect” if the ratio of the radii of the inner and outer cylinders is too small. For parallel wire probes, most of the measurement sensitivity is close to the wires if the wire diameter is too small compared to the spacing between them. This can cause significant errors if there is an air gap close to the wires or soil around the wires has been compacted by the process of inserting them into the soil.

A micromechanical approach to inelastic behaviour of metals

Abstract This paper describes micromechanical constitutive equations for polycrystals and shows their modelling capabilities. The field of interest of the model concerns monotomic and cyclic multiaxial loadings, with the assumption of small perturbation. First we discuss the type of physical variables that have to be introduced on a microscale. The case of FCC metals is especially investigated, so that the localization process includes the evaluation of stresses and strains for each grain and each octahedral slip system. The stress-strain relations are written on a microscale for each slip system. All the elementary contributions are then summed up to obtain the macroscopic strain rate. Two types of applications are shown. One deals with the representation of yield surfaces for a 2024 aluminium alloy tested at room temperature. A good agreement is found between numerical simulations and experiments for the initial and the subsequent surfaces after tension-torsion loading paths. The second identification is made on a 316L stainless steel, also tested in tension-torsion at room temperature. For this last case, the model is able to account for the memory effect and the supplementary hardening related with the non-proportionality of the loading path.

A Theoretical Study on Free-Surface Flow around Slowly Moving Full Hull Forms in Short Waves

This paper describes a theory on the free-surface flow slowly moving full hull forms in short waves. Under the assumption of small perturbation over the double body flow, a new function is derived which satisfies the free-surface condition including the effects of deformed flow due to the presence of the ship hull. By using the function, an expression of the velocity potential is introduced. In order to obtain the basic wave resistance feature of the present theory, calculations are made of the wave pattern generated by a pulsating source moving with constant speed below the free-surface.

A Rankine Panel Method to Calculate Unsteady Ship Hydrodynamic Forces

A panel method to calculate unsteady ship hydrodynamic forces is introduced. First, formulations were made for both steady and unsteady free-surface conditions under the assumption of small perturbation over the double body flow around a ship. A newly derived free-surface condition for unsteady motion makes a theoretical pair with Dawson's steady linearized free-surface condition. Next, a Rankine panel method was applied to solve the equations based on the present unsteady free-surface condition. For radiation condition of waves a numerical damping was introduced into the free-surface condition. Calculations were made of the unsteady hydrodynamic forces such as added mass, damping and wave exciting forces for a two-dimensional submerged cylinder and an ellipsoidal ship hull form. By comparing with other calculations and experiments it was shown that the present numerical method is useful for better understanding of the unsteady free-surface flow problems.

A computational procedure based on eigenvalue sensitivity theory applicable to linear system design

A digital computer based method for generating the sensitivities of the eigenvalues of a linear, time invariant, lumped parameter dynamic system to perturbations in its design parameters is described. The equations defining the particular system being considered must be supplied in a subroutine, and the sensitivity results are displayed in graphical form. Provision is made for treating one parameter in a special way appropriate to problems in vehicle and rotor dynamics, in which context satisfactory operation over a wide speed range is desirable, and for systematically combining parameter changes using the sensitivities generated to modify the eigenvalues in a desired manner. Although the sensitivity theory itself is based on the assumption of small design parameter perturbations, relatively large changes can be accommodated by combining small changes. It is claimed that there are many instances in which the use of the facility will be preferred to the commonly practised repetitive numerical generation of eigenvalues for different sets of design parameters.

Numerical simulation of scattering from simple and composite random surfaces

Numerical simulation studies of backscattering from one-dimensional, statistically known, perfectly conducting random surfaces are carried out to examine (1) the scattering characteristics in the intermediate-frequency region, where neither the low- nor the high-frequency approximation is applicable, (2) the validity of the two-scale concept in rough-surface scattering, and (3) applicability of the wavelength filtering concept to simple and composite surfaces. It is found that in the intermediate-frequency region the angular scattering curves for both the horizontal and vertical polarizations decrease at a slower rate than that predicted by a first-order small perturbation solution and that the Kirchhoff solution always lies between the two polarizations. In addition, the level difference between the horizontal and vertical polarization scattering coefficients is smaller than that of the perturbation solution. The two-scale concept has no general applicability. Its application is restricted to two-scale surfaces where one scale satisfies the Kirchhoff approximation and the other the small perturbation assumptions such that the two scales dominate scattering in different angular regions. The concept of wavelength filtering is applicable to simple surfaces or to each component of a composite surface in the low-frequency region where scattering is dominated by a narrow band of surface frequency components but loses its validity gradually as we approach the high-frequency limit where scattering is determined by the surface slope distribution. It is not the explanation of why the surface components of a composite surface may dominate scattering over different angular regions.

AN EXTENSION OF THE BORN INVERSION METHOD TO A DEPTH DEPENDENT REFERENCE PROFILE*

AbstractAn extension of the multidimensional Born inversion technique for acoustic waves is described. In earlier work, a perturbation in reference sound velocity was determined by assuming that the reference velocity was constant. In this extension, we allow the reference velocity to be a function of the depth variable z. The output of this method is a high‐frequency bandlimited reflectivity function of the subsurface. The reflectivity function is an array of bandlimited singular functions scaled by the normal reflection strength. Each singular function is a Dirac delta function of a scalar argument which measures distance normal to a reflecting interface. Thus, the reflectivity function is an indicator map of subsurface reflectors equivalent to the map produced by migration. In addition to the assumption of small perturbation, the method requires that the reflection data reside in the high frequency regime in a well‐defined sense.The method is based on the derivation of an integral equation for the perturbation in sound velocity from a known reference velocity. When the reference velocity is constant, the integral equation admits an analytic solution as a multifold integral of the reflection data. Further high frequency asymptotic analysis simplifies this integral considerably and leads to an extremely efficient numerical algorithm for computing the reflectivity function. The development of a computer code to implement this constant‐reference‐velocity solution is published elsewhere.For a reference velocity c(z) we can no longer invert the integral equation exactly. However, we can write down an asymptotic high‐frequency approximation for the kernal of the integral equation and an asymptotic solution for the perturbation. The computer implementation of this result is designed along the same lines as the code for constant background velocity. In tests the total processing time for this algorithm with depth‐dependent background velocity is usually considerably less than that required by a standard Kirchhoff migration algorithm. The method is implemented as a migration technique and compared with alternative migration algorithms on the flanks of the salt dome.

Minimum probability of electron-hole pair generation by blackbody radiation

Using the small-perturbation assumption that all the useful work is in the form of electron-hole pairs when photons from a blackbody radiation source are incident on a quantum device, we derive the minimum probability of generation of one or more pairs of electron holes. Using the Planck radiation formula for the solar spectrum at 6000 K and for the quantum device at 300 K, this minimum probability is about 0.8 at low photon frequency and approaches the limiting value of 0.95 at high frequency. Its relation to maximum efficiency of a quantum device is discussed.

Stability Threshold and Steady-State Response of Noncontacting Coned-Face Seals

The dynamic behavior of a noncontacting coned-face seal is analyzed. Stiffness and damping properties of the fluid film and flexible support including elastomeric secondary seal are fully accounted for. Stability threshold and steady-state response in the presence of rotor axial runout and assembly misalignment are investigated. An expression is provided for the critical speed above which the seal becomes dynamically unstable. For stable operation, the relative misalignment between the mating faces is given as a function of rotor runout, assembly misalignment, design parameters, and operation conditions. An expression is provided for the critical rotor runout above which the seal will fail due to face contact. Although the analysis is based on small perturbation assumption, it is shown to be valid in many practical cases.

A Second-Order Theory for the Potential Flow About Thin Hydrofoils

This research addresses the potential flow about a thin two-dimensional hydrofoil moving with constant velocity at a fixed depth beneath a free surface. The thickness-to-chord ratio of the hydrofoil and disturbances to the free stream are assumed to be small. These small perturbation assumptions are used to produce first-and second-order subproblems structured to provide consistent approximations to boundary conditions on the body and the free surface. Nonlinear corrections to the free-surface boundary condition are included at second order. Each subproblem is solved by a distribution of sources and vortices on the chord line and doublets on the free surface. After analytic determination of source and doublet strengths, a singular integral equation for the vortex strength is derived. This integral equation is reduced to a Fredholm integral equation which is solved numerically. Lift, wave drag, and free-surface shape are calculated for a flat plate and a Joukowski hydrofoil. The importance of free-surface effects relative to body effects is examined by a parametric variation of Froude number and depth of submergence.

Unsteady response of rectangular wings in spanwise uniform shear flow

In engineering applications, it is important to estimate aerodynamic forces on wings in nonuniform flow. Wakes produced by objects ahead of wings frequently affect the performance of airplanes, turbomachines, etc. In this paper, a theory of wings oscillating in shear flow is presented. The whole flowfield is assumed to be inviscid and incompressible, and, furthermore, the shear flow velocity is assumed to vary linearly along the wing span. For this problem, lifting surface theory is applied. An integral equation, which is similar to that for potential flow, is derived under the assumption of small perturbations, arid then solved numerically by the mode function method. Generalized forces, Q^, which can be easily related to unsteady lift forces and moments, are obtained; moreover, response functions of wings to sinusoidal heaving oscillations and a sinusoidal gust are calculated for three different amounts of shear flow. The results are compared for several frequencies and show that the effect of shear is not large except in the limiting case. However, the generalized forces and gust response functions are affected by .the shear to some extent, even when the degree of shear is moderate. The effect increases as the frequency becomes larger.

Asymmetric Distortion Generation in a Variable Height Annulus

POROUS screens were placed in the compressor intake to create quasisteady pressure maldistributed inlet flows. This study was aimed at creating a single-lobe circumferential sine-wave velocity profile with no radial components at the compressor inlet. A method relating screen blockage to predicted velocity levels was devised. The method yields velocity levels close to those for which the screen was designed. Contents Quasisteady pressure maldistributed compressor inlet flows are normally created during testing by the inclusion of porous screens in the compressor intake, but interactions between a row of compressor rotor blades and upstream placed screens in such experiments can alter the shape of the generated distortion.l Suppression of this effect may be realized by remote positioning of the screen upstream of the compressor leading-edge plane. Often, an intake may not be a constant annulus, so flow differences between the compressor leadingedge plane and the plane of the screen call for differences in the screen design whose generated distortion is itself modified in passing through the annulus to the compressor. The aim of this study was to present to a compressor inlet a single-lobe circumferential sine-wave velocity profile with no radial components, the distortion being generated remotely upstream in a duct dissimilar to the annulus at the compressor intake. Based upon small perturbation assumptions, a method of screen design is proposed which superimposes a generated distortion upon the undistorted potential flow solution of the developed annulus. The example used was a bellmouth and nose-bullet sited coaxially close to the compressor leadingedge plane and separated only by a short length of the constant height annulus. The screen was set at the bellmouth intake, perpendicular to the axial direction. While this example used symmetrical ducting, the principles involved could be applied to a nonsymmetrical installation.

Bond Graphs for Vehicle Dynamics

SUMMARY For the purposes of analyzing the stability and control properties of aircraft, automobiles, and ships, it is convenient to idealize the vehicle as a rigid body and to use a body-centered coordinate frame. A six-port bond graph is first developed which represents the nonlinear dynamics of a rigid body in a coordinate system rotating with the body. It is then shown how the graph simplifies if only lateral or longitudinal dynamics are studied. Finally, bond graphs for the type of stability studies in which the assumption of small perturbations permits linearization are shown for elementary models of automobiles and aircraft.

Radiation-convection interaction for a laterally bounded flow past a hot plate

Radiation-convection interaction for a laterally bounded flow past a heated plate is investigated, whereby the influence of two-dimensional radiation is emphasized. Due to the high anisotropy of the radiation near the leading edge of the plate the ordinary, two-dimensional differential approximation fails in the vicinity of the edge. Therefore Olfe's modified differential approximation, which separately accounts for the external wall radiation, is employed. The equations for a gray gas are linearized under the assumption of small temperature perturbations and are solved by means of the Wiener-Hopf technique. The results are compared with several approximate solutions. Among other results it is shown that the one-dimensional “radiation-layer” model of Cess is limited to weak radiation. For intermediate and strong radiation a distinct precursor is observed.