Approximate Perturbation Solution Research Articles

VIBRATIONS OF A VERTICAL BEAM ROTATING WITH VARIABLE ANGULAR VELOCITY

An Euler-Bernoulli beam in vertical position rotating about its symmetry axis along its length is considered. The angular velocity is assumed to have small fluctuations about a constant mean velocity. The partial differential equation of motion is derived first. The equation is cast into a non-dimensional form. The natural frequencies are calculated for the pinned-pinned case. Principle parametric resonances such that the fluctuation frequency being close to two times one of the natural frequencies are considered. By employment of the Method of Multiple Scales, an approximate perturbation solution is found. The frequency response diagrams are drawn and the bifurcation points for transition from the trivial solution to the non-trivial solution are calculated. The conditions for which such resonances occur are exploited in the numerical results.

Exact Algebraic Solution of an Optimal Double-Mass Dynamic Vibration Absorber Attached to a Damped Primary System

AbstractThis article presents exact algebraic solutions to optimization problems of a double-mass dynamic vibration absorber (DVA) attached to a viscous damped primary system. The series-type double-mass DVA was optimized using three optimization criteria (the H∞ optimization, H2 optimization, and stability maximization criteria), and exact algebraic solutions were successfully obtained for all of them. It is extremely difficult to optimize DVAs when there is damping in the primary system. Even in the optimization of the simpler single-mass DVA, exact solutions have been obtained only for the H2 optimization and stability maximization criteria. For H∞ optimization, only numerical solutions and an approximate perturbation solution have been obtained. Regarding double-mass DVAs, an exact algebraic solution could not be obtained in this study in the case where a parallel-type DVA is attached to the damped primary system. For the series-type double-mass DVA, which was the focus of the present study, an exact algebraic solution was obtained for the force excitation system, in which the disturbance force acts directly on the primary mass; however, an algebraic solution was not obtained for the motion excitation system, in which the foundation of the system is subjected to a periodic displacement. Because all actual vibration systems involve damping, the results obtained in this study are expected to be useful in the design of actual DVAs. Furthermore, it is a great surprise that an exact algebraic solution exists even for such complex optimization problems of a linear vibration system.

Compressible liquid flow in nano- or micro-sized circular tubes considering wall–liquid Lifshitz–van der Waals interaction

Although nano- and micro-scale phenomena for fluid flows are ubiquitous in tight oil reservoirs or in nano- or micro-sized channels, the mechanisms behind them remain unclear. In this study, we consider the wall–liquid interaction to investigate the flow mechanisms behind a compressible liquid flow in nano- or micro-sized circular tubes. We assume that the liquid is attracted by the wall surface primarily by the Lifshitz–van der Waals (LW) force, whereas electrostatic forces are negligible. The long-range LW force is thus introduced into the Navier–Stokes equations. The nonlinear equations of motion are decoupled by using the hydrodynamic vorticity-stream functions, from which an approximate analytical perturbation solution is obtained. The proposed model considers the LW force and liquid compressibility to obtain the velocity and pressure fields, which are consistent with experimentally observed micro-size effects. A smaller tube radius implies smaller dimensionless velocity, and when the tube radius decreases to a certain radius Rm, a fluid no longer flows, where Rm is the lower limit of the movable-fluid radius. The radius Rm is calculated, and the results are consistent with previous experimental results. These results reveal that micro-size effects are caused by liquid compressibility and wall–liquid interactions, such as the LW force, for a liquid flowing in nano- or micro-sized channels or pores. The attractive LW force enhances the flow’s radial resistance, and the liquid compressibility transmits the radial resistance to the streaming direction via volume deformation, thereby decreasing the streaming velocity.

An analytical technique to find approximate solutions of nonlinear damped oscillatory systems

Combining Krylov–Bogoliubov–Mitropolskii (KBM) and harmonic balance methods, an analytical technique is presented to determine approximate solutions of nonlinear oscillatory systems with damping. The first approximate perturbation solutions in which the unperturbed solutions contain two harmonic terms agree with numerical solutions nicely even if the damping force is significant. With suitable examples it has been shown that the combination of classical KBM and harmonic balance methods sometimes fails to measure satisfactory results; but the combination of extended KBM method (by Popov) and harmonic balance method always give the desired results. The method is illustrated by several examples and the solutions are compared to some existing solutions.

Tide-induced seawater–groundwater circulation in a multi-layered coastal leaky aquifer system

Mean groundwater levels of a multi-layered coastal leaky aquifer system are considered. The system consists of an unconfined aquifer, a confined aquifer and a semi-permeable layer between them. Both exact asymptotic solutions and approximate perturbation solutions are derived for multi-sinusoidal-component sea tide. At inland places far from the coastline, the perturbation solutions show a good agreement with the exact asymptotic solutions. Due to the watertable-dependent transmissivity of the unconfined aquifer, the mean groundwater levels of the aquifer system stand considerably above the mean sea level even in the absence of net inland recharge of groundwater and rainfall. These lead to landward positive gradients of both the mean watertable and mean head in the region near the coastline, which consequently results in a seawater–groundwater cycle. Seawater is pumped into the unconfined aquifer by the sea tide and divided into two parts. One part returns to the sea driven by the mean watertable gradient. The rest part leaks into the confined aquifer through the semipermeable layer, and returns to the sea through the confined aquifer driven by the mean head gradient. The total discharge through the confined aquifer is significant for coastal leaky aquifer system with typical parameter values. This seawater–groundwater cycle has impacts on better understanding of submarine groundwater discharge and exchange of various chemicals such as nutrients and contaminants in coastal areas. If the observed mean water levels in coastal areas are used for estimating the net inland recharge, the enhancing processes of sea tide on the mean groundwater levels should be taken into account. Otherwise, the net inland recharge will be overestimated.

Scale-Dependent Dispersion from a Pit

It is fairly well accepted that contaminant transport in aquifers can be modelled accurately only if dispersion coefficients are allowed to increase with distance downstream. However, this also increases the difficulty of obtaining exact solutions of the contaminant transport equation. Therefore, singular perturbation methods are used to obtain a relatively simple approximate solution for scale-dependent dispersion from a pit located in a region of uniform flow. Comparison of exact and approximate perturbation solutions for two simpler but related problems suggests that the perturbation approach used here is likely to give sufficient accuracy for many, if not most, applications.

Multicompartment urea kinetics in well-dialyzed children

We have reported catch-up growth with hemodialysis (HD) of approximately 15 hours/week. Without an equilibrated post-treatment blood urea nitrogen, the variable-volume single-pool (VVSP) model will not account for urea rebound, inflating the estimated HD dose (K(d)t/V). A two-pool model (FVDP) predicts rebound, but requires fixed compartment volumes for the equations to be solvable in closed form, also inflating K(d)t/V. We developed an approximate perturbation solution (WKB method) to a variable volume, two-pool (VVDP) model. Estimated model parameters were compared with the results of equilibrated kinetic studies using measured clearance K(d) (N = 17). Once the model was validated, we re-analyzed 292 kinetic studies from our earlier cohort, which was considered well-dialyzed on the basis of growth rates (N = 12, mean annual change in height standard deviation score +0.31, mean follow-up of 26 months). For the VVSP, FVDP, and VVDP models, respectively, the mean errors were (1) K(d)t/V, 0.22 +/- 0.07, 0.29 +/- 0.17, 0.06 +/- 0.07 (ANOVA, P < 0.001); (2) urea distribution volume vol/wt (%), -8.2 +/- 4.2, -9.1 +/- 3.0, -2.2 +/- 3.6 (P < 0.001). Sequential studies confirmed reproducibility, with a coefficient of variation < or = 5%. In the earlier cohort, a comparison of the VVSP and VVDP models yielded the following: (1) K(d)t/V, 1.91 +/- 0.35 vs. 1.76 +/- 0.33 (P < 0.001); (2) normalized protein catabolic rate (nPCR, g/kg/day), 1.56 +/- 0.39 vs. 1.52 +/- 0.38 (P < 0.001); and (3) K(d) (whole blood, mL/kg/min), 4.8 +/- 0.9 vs. 4.4 +/- 0.8 (P < 0.001). This VVDP model yields reliable estimates of K(d)t/V and other kinetic parameters using standard blood urea nitrogen sampling. Analysis of patients previously characterized as well-dialyzed on the basis of growth rates clarifies the HD dose needed to sustain normal growth.

Simple Nonlinear Rainfall-Runoff Model

A simple nonlinear conceptual model for simulating Hortonian rainfall-runoff phenomena is presented. The model is based on the concept of a nonlinear storage outflow relationship and assumes a nonlinear loss module. The abstractions can take the form of a constant or an exponential decreasing rate. The analytical model is derived from an approximate perturbation solution of the governing nonlinear ordinary differential equation and relates in an explicit fashion the runoff to the rainfall, the abstractions, and the watershed parameters. The model is function of few parameters that can be estimated efficiently using optimization techniques and nonlinear least-squares regression. The model has been applied to a very small semiarid watershed with considerable success. The proposed model can be considered as an improvement over the classical linear reservoir theory, and an attractive and useful substitute to the linear reservoir components in more complex conceptual rainfall-runoff models.

Radiation due to a convex curvature discontinuity of a dielectric-coated perfect conductor

Surface waves radiate energy at discontinuities in the curvature of the guiding structure. By reciprocity, surface waves will be excited by plane waves incident upon such a discontinuity. Here, the problem of the radiation of a surface wave on a flat dielectric-coated perfect conductor incident upon an abrupt change to a dielectric-coated cylindrical conductor with a large radius of curvature is considered. The problem is formulated as an integral equation over the aperture of the discontinuity. Since the change in curvature is modest, an approximate perturbation solution to the integral equation is derived and the radiated field due to the discontinuity is found. This radiated field reduces to published results for an impedance surface approximation when that approximation is valid. The problem of mode conversion and associated radiation near higher mode cutoffs is also studied. It is found that near mode cutoffs, the higher order mode dominates the radiation pattern and causes the overall radiation pattern due to the discontinuity in curvature to be narrow and end fire. Away from cutoff, when all of the propagating bound modes are more tightly bound to the surface, the radiation pattern is less narrow and less end fire. For very tightly bound modes the pattern is nearly uniform. For dielectrics characterized by small permitivities, the changes in radiation pattern should be measurable.

A Geostatistical Inverse Method for Variably Saturated Flow in the Vadose Zone

A geostatistical inverse technique utilizing both primary and secondary information is developed to estimate conditional means of unsaturated hydraulic conductivity parameters (saturated hydraulic conductivity and pore size distribution parameters) in the vadose zone. Measurements of saturated hydraulic conductivity and pore size distribution parameters are considered as the primary information, while measurements of steady state flow processes (soil‐water pressure head and degree of saturation) are regarded as the secondary information. This inverse approach relies on the classical linear predictor (cokriging) theory and takes the advantage of the spatial cross correlation between the soil‐water pressure head and each of the following: degree of saturation, saturated hydraulic conductivity, and a pore size distribution parameter. Using an approximate perturbation solution for steady, variably saturated flow under general boundary conditions, the cross covariances between the primary and secondary information are derived. The approximate solution is formulated on the basis of a first‐order Taylor series expansion of a discretized finite element equation. The sensitivity matrix in the solution is evaluated by an adjoint state sensitivity approach for flow in heterogeneous media under variably saturated conditions. Through several numerical examples the inverse model demonstrates its ability to improve the estimates of the spatial distribution of saturated hydraulic conductivity and pore size distribution parameters using the secondary information.

Analytical models for relative motion under constant thrust

A general formulation for relative motion is presented allowing for arbitrary perturbing or thrust forces on each of the two satellites. Exact as well as approximate perturbation solutions for the relative motion under constant radial or circumferential forces acting on the subsatellite are established. The validity and usefulness of these solutions is assessed for a few realistic applications related to EURECA rendezvous manoeuvering with the Shuttle. The results are of general Interest for the fast calculation of relative subsatellite motion under thrust forces.

Nonlinear fluid slosh coupled to the dynamics of a spacecraft

The dynamics of a linear spacecraft mode coupled to the nonlinear low-gravity slosh of a fluid in a cylindrical tank is investigated. Coupled, nonlinear equations of motion for the fluid-spacecraft system are derived through an assumed mode Lagrangian method. Unlike a linear model, this nonlinear model retains two fundamental slosh modes and three secondary slosh modes. An approximate perturbation solution of the equations of motion indicates that the nonlinear coupled system response involves fluid-spacecraft modal resonances not predicted by either a linear, or a nonlinear, uncoupled slosh analysis. An experiment was developed to verify and complement this analysis. Scale model fluid tanks were coupled to an electromechanical analog for the second-order oscillatory spacecraft mode. Moderate and low gravity were simulated in 1 g using capillary scale models, and zero gravity was simulated in parabolic flight tests on the NASA KC-135 Reduced Gravity Test Facility. The experimental results substantiated the analytical predictions. The dependence of the coupled nonlinear response on the coupled system parameters and the gravity level is illustrated and discussed.

Steady laminar flow through a twisted pipe of elliptical cross-section

Previous work by the author on a formal series method of solution for flows in pipes with slowly varying geometries which provides a basis for approximate perturbation solutions is applied to the specific example of flow in a slowly twisted pipe of elliptical cross-section. With the aid of the algebraic and symbolic manipulation programming language REDUCE the calculations to low order have been carried out to reveal some purely three-dimensional effects.

Contact—angle evaluation from small sessile drops: A simplified perturbation approach

The analytically insoluble second-order differential equation describing the profile of axisymmetric liquid—fluid interfaces in a gravitational field is of interest for both interfacial tension and contactangle determination. A recent paper by the author described an approximate perturbation solution to the equation for sufficiently small sessile (or pendant) drops of contact angle, θ, <90°. This note describes a method to simplify further the perturbation solution and thus render its application easier in contact-angle evaluation. The range of validity of the new equation is investigated by comparison with accurate computer-calculated data.

Reduced equilibrium equations for perfectly elastic materials

Sufficient conditions are given on the coordinate systems which enable reduced equilibrium equations to be derived for perfectly elastic materials involving deformations which depend in an essential way only on two of the three coordinates. Reduced equilibrium equations given previously for plane and axially symmetric deformations are special cases of the equations given here. These equations considerably reduce the calculations involved in investigating possible solutions of finite elasticity, either exact semi-inverse solutions or approximate perturbation solutions. Moreover a formula for the pressure function appearing in the reduced equilibrium equations is given which relates to the corresponding pressure function associated with the inverse deformation. This formula is similar to one given previously for fully three dimensional deformations.

On the matching of two optical waveguides

The problem of matching end to end of two semi‐infinite, single or multimode optical dielectric waveguides is formulated. When the waveguides' properties are different, or their axes do not coincide, reflection and radiation occur from the point of discontinuity, resulting in transmission losses. The reflection and transmission matrices and the radiation coefficients are found to satisfy certain singular integral equations. Approximate (perturbation) solutions are obtained for cases of small discontinuities in the waveguides' properties or axial alignment.

The dispersed flow model for a biological reactor as applied to the activated sludge process

AbstractA theoretical analysis of a dispersed flow biological reactor is presented in which Monod growth kinetics are used to describe the substrate utilization rate and the bacterial growth rate. Different bacterial and substrate dispersion coefficients are taken into account. Approximate perturbation solutions are established for two limiting situations, when the inverse Peclet number ϵ is very small and when it is very large. In the former case the perturbation analysis is singular. These perturbation solutions are compared with the solutions of the numerically integrated equations over a range of parameters corresponding to actual activated sludge systems. The application to the activated sludge system is discussed.

Statistical theory of effective electrical, thermal, and magnetic properties of random heterogeneous materials. V. One− and two−dimensional systems

A perturbation theory is developed for the effective permittivity of one− and two−dimensional random heterogeneous materials that are statistically homogeneous. Closely following the formulations for three−dimensional systems presented in Papers I−IV in this series of work, we derive the formal perturbation solutions for two−dimensional systems, evaluate the second−order and third−order terms for cell materials, and determine upper and lower bounds of the effective permittivity. Here statistical isotropy is not necessarily required. Several approximate perturbation solutions for completely random systems (which are statistically isotropic) are obtained by summing some selected partial series of the perturbation expansion up to an infinite order and numerical results are illustrated. We analyze validity of approximations by means of diagram representation of the perturbation series. The effective−medium theory serves as a good approximation for two−dimensional systems and gives the critical percolation concentration correctly. For one−dimensional materials, the effective−medium approximation turns out to be an exact solution.

High flux solid-liquid mass transfer

The effect of large concentration gradients upon the rate of mass transfer from a rotating disc has been measured experimentally and compared with theory. Measured rates were found to agree with the numerical solutions of the variable property, finite interfacial velocity diffusion equation. The data were also compared with approximate perturbation and integral solutions of the diffusion equation. The systems investigated were benzoic acid, potassium bromide, sucrose, and copper sulfate in water. Variations in density, viscosity and diffusivity and the existence of a diffusion-induced interfacial velocity can result in differences between the constant property, zero interfacial velocity and the actual Sherwood numbers as large as a factor of two.