Simple Approximate Formula Research Articles

Synchronization of oscillators coupled through narrow-band networks

The ability of two coupled oscillators to synchronize depends critically on the coupling network. Previous analyses have accurately predicted the performance of quasi-optical microwave oscillator arrays for both weak and strong coupling, but have been limited to coupling networks with bandwidths considerably larger than the locking bandwidths of the oscillators. In this paper, the authors develop a method for deriving a suitable system of nonlinear differential equations describing the oscillator amplitude and phase dynamics using a generalization of Kurokawa's method. The method is applied to the case of two Van der Pol oscillators coupled through a resonant network for a wide range of coupling strengths and bandwidths. Simple approximate formulas are developed for the size of the frequency locking region as functions of the basic circuit parameters.

診断領域Ｘ線の実効エネルギー推定法

Because X-ray exposure doses to patients during X-ray diagnoses have been increasing with recent advances in medical technology, it is important that optimum control of the radiation dose be maintained during diagnoses. For an evaluation of an exposure dose, the effective energy of the X-ray must be determined, but this is difficult to accomplish during the diagnosis. Here we propose a new method to estimate the effective energy of an X-ray. The magnitude of energy released from an X-ray generator (2 peaks, 12 peaks, inverter, and constant potential) depends on various parameters, including tube voltage, tube current, tube voltage waveform, and total filtration of the X-ray tube. Therefore the measurement of an X-ray's effective energy was conducted by the half-value layer measurement method, which changes the values of these parameters. The data obtained by this method were analyzed to clarify the relationships between X-ray effective energy and the respective parameters. It was thus demonstrated that these relationships could be expressed by a simple linear approximation formula. For the calculation of X-ray effective energy by use of this approximation formula, errors were found to be within a range of -2.11% to 10.4%. Therefore, this method is considered usable for an accurate estimation of an X-ray's effective energy without the need for its direct determination during diagnosis.

Limiting State of Transversely Isotropic Plates with Surface and Internal Cracks

We propose a method for the analysis of the limiting state of transversely isotropic plates with surface and internal cracks and obtain a new simple approximate formula for the evaluation of the stress intensity factors. The numerical calculations are based on the improved version of the analysis of stress-strain states in defect-free transversely isotropic plates. The failure is initiated as soon the damage ratio near the crack front in the direction of crack propagation attains a certain critical value.

Energy release during disk accretion onto a rapidly rotating neutron star

The energy release Ls on the surface of a neutron star (NS) with a weak magnetic field and the energy release Ld in the surrounding accretion disk depend on two independent parameters that determine its state (for example, mass M and cyclic rotation frequency f) and is proportional to the accretion rate. We derive simple approximation formulas illustrating the dependence of the efficiency of energy release in an extended disk and in a boundary layer near the NS surface on the frequency and sense of rotation for various NS equations of state. Such formulas are obtained for the quadrupole moment of a NS, for a gap between its surface and a marginally stable orbit, for the rotation frequency in an equatorial Keplerian orbit and in the marginally stable circular orbit, and for the rate of NS spinup via disk accretion. In the case of NS and disk counterrotation, the energy release during accretion can reach \(0.67\dot Mc^2 \). The sense of NS rotation is a factor that strongly affects the observed ratio of nuclear energy release during bursts to gravitational energy release between bursts in X-ray bursters. The possible existence of binary systems with NS and disk counterrotation in the Galaxy is discussed. Based on the static criterion for stability, we present a method of constructing the dependence of gravitational mass M on Kerr rotation parameter j and on total baryon mass (rest mass) m for a rigidly rotating neutron star. We show that all global NS characteristics can be expressed in terms of the function M(j, m) and its derivatives. We determine parameters of the equatorial circular orbit and the marginally stable orbit by using M(j, m) and an exact solution of the Einstein equations in a vacuum, which includes the following three parameters: gravitational mass M, angular momentum J, and quadrupole moment Ф2. Depending on Ф2, this solution can also be interpreted as a solution that describes the field of either two Kerr black holes or two Kerr disks.

Genetic algorithms applied to the selection of factors in principal component regression

Using principal component regression (PCR) as a multivariate calibration tool, always brings up the question what subset of factors, i.e. principal components (PCs) gives the best calibration model. Normally factor selection is based on deterministic methods like top–down procedures, forward–backward-stepwise variable selection or correlated principal component regression (CPCR). In contrast to this, we applied a stochastic method, i.e. a genetic algorithm (GA) for factor selection in this paper. A new kind of fitness function was applied which combined the prediction error of the calibration and an independent validation set. The performance of eigenvalue and correlation ranking was compared. A general statistical criterion for judging the significance of differences between individual calibration models is introduced. In this context it could be shown that for the uncertainties of the standard deviations representing the prediction errors a very simple approximation formula holds which only includes the number of standards. For the current applications it is shown that the GA gives a result very close to CPCR-solutions.

Muonium and muonium-like systems formation in spur model

Track electron kinetics is shown to be important for the correct understanding of μSR experiments. The quasi-neutrality restoring time for the region of muon or meso-ion location is investigated theoretically and numerically in different models relating to positive and negative muons. Simple approximate formulae for numerical results are presented. This time is found to be much less than the muon's lifetime for light gases and Ne at a liquid phase pressure. The foregoing makes it possible that meso-ion and electron interaction leads to the formation of Mu or muonium-like systems.

A simple interpolation formula for the spectraofpower-law and log potentials

Non-relativistic potential models are considered of the pure power V(r) = sgn(q) r q and logarithmic V(r) = ln (r) types. It is shown that, fromthe spectral viewpoint, these potentials are actually in a single family.The log spectra can be obtained from the power spectra by the limit q→0 taken in a smooth representation Pnℓ(q) for theeigenvalues Enℓ(q). A simple approximation formula is developedwhich yields the first 30 eigenvalues withan error less than0.04%.

Resonant frequency and bandwidth of a superstrate-loaded rectangular patch on a uniaxial anisotropic substrate

In this paper, the effects of both uniaxial anisotropy in the substrate and isotropic superstrate loading on the resonant frequency and bandwidth of a rectangular microstrip patch in a substrate–superstrate configuration are investigated. The problem is rigorously formulated using an integral equation, the kernel of which is the full-wave spectral domain dyadic Green's function for multilayer dielectric substrates. The proposed method for determining the Green's function leads to a concise form of this, diagonalized in the (TM,TE) representation. Using Galerkin's moment method to solve the integral equation, the complex resonance frequencies for the TM01 mode are studied with sinusoidal basis functions. The improper double integrals of the moment method matrix are efficiently calculated by means of a proper choice of the path of integration in the complex wavenumber plane, the upper bound of truncation, and the method of quadrature. For an isotropic substrate, it is demonstrated that the bandwidth decreases with increasing ratio of superstrate-to-substrate thickness for high permittivity and low thickness of superstrate. Also, we show that the resonant frequency and bandwidth are highly dependent on the permittivity variations along the optical axis. Simple approximate formula for the resonant frequency is derived. Other theoretical results obtained show that the resonant frequency decreases monotonically with increasing superstrate thickness, the decrease being greater for high permittivity loading and negative uniaxial anisotropy of the substrate. Thin superstrate with high permittivity together with negative uniaxial anisotropy of the substrate are shown to be the favourable conditions for severe degradation of the half-power bandwidth of the antenna. Copyright © 2000 John Wiley & Sons, Ltd.

Approximate Fourier coefficients for gapped magnetic recording heads

An investigation is presented of the number of exact Fourier terms needed to evaluate the field components of a ring head, a slotted plane, and planar heads of various dimension ratios to specified accuracy. Simple approximate formulas for the coefficients of a ring head and a slotted plane are given, and two approximations to the coefficients of a planar head of given dimensions are provided. The accuracy of the field components and the spectral response function, each evaluated with the approximate coefficients, is used to assess the usefulness of these approximations. Both of the approximations for the coefficients of planar heads lead to excellent estimates of the field components.

A simple formula for the ground resistance calculation

Simple approximate formulas for the ground resistance and the potential distribution that have quite good accuracy are presented. The use of these formulas can simplify and make faster the calculation of ground path coupling.

STRESS DECAY IN AN ORTHOTROPIC HALF-PLANE UNDER LOCAL SELF-EQUILIBRATED EDGE LOADING

An exact solution is presented for the stresses in a specially orthotropic half-plane under a localized, symmetric, self-equilibrated combination of uniform normal edge loadings. “Smeared” orthotropic material properties are used to represent a broad range of balanced angle-ply graphite-epoxy laminates. Expressions for the three stresses in the half-plane are presented for isotropic material and for two orthotropic material families that contain many of the practical laminates. Particular attention Is given to the diffusion of the edge loading into the half-space and to the shear stresses caused by the discontinuous edge loading, both of which are strongly influenced by material properties. Of special interest to designers of composite structures may be a simple approximate formula, in terms of material properties, which is proposed for a lower bound on the normalized 90%-decay distance for the normal stress. Also, the results serve as further illustration of the fact that the meaningful application of Salnt-Venant's principle to specially orthotropic materials requires knowledge of their elastic properties.

Simple formula for AM-detector transfer factor

A simple approximate formula for the AM-detector transfer factor, which is valid for small-signal modes as well as for large-signal ones is proposed. This formula allows one to model analytically (or numerically at the behavioural level) an AM-detector in the large-signal and small-signal modes taking into account the nonlinearity of its transfer factor. Formula validation is made by comparison with measurements and PSPICE simulation data.

CQC法におけるモード相関係数の簡便な近似式とその適用例

For classically damped systems subjected to a white-noise input, a simple approximation formula of modal correlation coefficient in CQC method is proposed and its application examples are shown. This proposed formula has both enough accuracy and enough simplicity for practical use. Its accuracy is very high if both of the damping constants of two modes are less than 0.1. Compared with the exact solutions, this formula is greatly simplified. For example, this is common to all kinds of response values and has a simple expression directly reflecting the fundamental properties of the modal correlation. Thus, the effects of the modal correlation can be evaluated very easily by this formula. Simple cases of the torsional coupling and the tuned mass damper are shown as such examples.

Hermite-Gaussian expansion for pulse propagation in strongly dispersion managed fibers

We represent a pulse in the strongly dispersion managed fiber as a linear superposition of Hermite-Gaussian harmonics, with the zeroth harmonic being a chirped Gaussian with periodically varying width. We obtain the same conditions for the stationary pulse propagation as were obtained earlier by the variational method. Moreover, we find a simple approximate formula for the pulse shape, which accounts for the numerically observed transition of that shape from a hyperbolic secant to the Gaussian. Finally, using the same approach, we systematically derive the equations for the evolution of a pulse under a general perturbation. This systematic derivation justifies the validity of similar equations obtained earlier from the conservation laws.

Padé approximants for the acoustical properties of rigid frame porous media with pore size distributions

Expressions for the viscosity correction function, and hence bulk complex impedance, density, compressibility, and propagation constant, are obtained for a rigid frame porous medium whose pores are prismatic with fixed cross-sectional shape, but of variable pore size distribution. The low- and high-frequency behavior of the viscosity correction function is derived for the particular case of a log-normal pore size distribution, in terms of coefficients which can, in general, be computed numerically, and are given here explicitly for the particular cases of pores of equilateral triangular, circular, and slitlike cross-section. Simple approximate formulae, based on two-point Padé approximants for the viscosity correction function are obtained, which avoid a requirement for numerical integration or evaluation of special functions, and their accuracy is illustrated and investigated for the three pore shapes already mentioned.

Acoustic resonances in the presence of radial fins in circular cylindrical waveguides

This paper presents a method, based on matched eigenfunction expansions, for the determination of the frequencies of the acoustic resonances that can occur in a circular cylindrical waveguide in the presence of a number of thin radial fins of finite length in the axial direction. The fins are evenly distributed around the guide and we find that there are significant differences in the possible modes depending on whether the number of fins is even or odd and also that the qualitative behaviour of the mode frequencies changes when the number of fins passes a certain critical value. Simple approximate formulas are derived for the low-frequency modes under the assumption that the number of fins is large. The implications of our results are discussed for the case when the circular cross-section is replaced by an annular one, a situation of practical importance in the design of turbomachinery.

Analytical solutions for rotational diffusion in the mean field potential: application to the theory of dielectric relaxation in nematic liquid crystals

A theory of dielectric relaxation in nematics is developed for a molecular dipole moment directed at an arbitrary angle to the molecular long axis. Both exact and simple approximate analytical formulae for the longitudinal and transverse components of the complex dielectric permittivity tensor are obtained for the non-inertial rotational Brownian motion of a molecule in the mean field potential of Maier and Saupe. It appears that both longitudinal and transverse relaxation processes are effectively described by two Debye type mechanisms with corresponding relaxation times and dielectric strengths expressed in terms of the order parameter. The generalization of the theory for an arbitrary axially symmetric mean field potential is given.

Transition to just‐in‐time purchasing

The transition from a traditional purchasing system to a JIT purchasing system can be a slow process or even unattainable, because of unreliable suppliers. The purchaser tries to co‐operate with the vendor, with the goal of receiving smaller, more frequent deliveries, on time, with the quality and quantity required. Often the vendor is ready to co‐operate, but is unable to fulfil these requirements. Provides simple models and methods to aid purchasers in this transition state. Gives simple approximate formulas for the minimum safety stock necessary to ensure the required service level of supply. Considers the case of random delays in shipments, random yield and uncertain demand, which are typical characteristics during the transition period. This safety stock depends on the order quantity and the number of shipments. Provides a simple method to find the order quantity, the number of shipments and safety stock, which minimize the joint total cost of the vendor and purchaser and ensure the required level of supply. Analyzes the savings provided by this method and the sensitivity of the models, in detail.

Diffraction of a plane wave by a narrow cone

The problem of diffraction of a plane scalar wave by a narrow cone is considered. The shape of the cone is arbitrary. The boundary condition is the Dirichlet or Neumann one. The wave scattered by the cone vertex arises as a result of the diffraction process. The subject of this paper is to calculate the wave amplitude. If the cone is narrow, it is possible to obtain simpler approximate formulas in comparison with Smyshlayev's one. The exactness of the approximate formulas is checked numerically. The etalon is a solution in explicit form in the axially symmetric case. The calculation shows that our formula is more exact in the case of the Dirichlet boundary condition than Felsen's formula. The approximate formula is a generalization of Felsen's one for circular cone to an arbitrary narrow cone in the case of the Neumann boundary condition. Bibliography: 6 titles.

Crossover condition in the coherent rotation model and the Preisach-type models

The crossover of the hysteresis branches in the classical coherent rotation model is analyzed. The analysis is extended when the truncated Stoner-Wohlfarth astroid is used. The physical explanation of the crossover occurrence is given. More accurate expressions of the anisotropy free energy density are required to better describe the behavior of the system when the angle between the easy axis and the magnetic moment of the particle is near 90/spl deg/. A simple approximate formula for the hysteresis loop branches, which are in good agreement with those obtained when the discussed corrections are made, is given. The effect of the use of the Stoner-Wohlfarth astroid in the calculus of the equilibrium state in some vector Preisach-type models is analyzed.