Single-term Approximation Research Articles

Water wave scattering by a pair of thin vertical barriers with submerged gaps

The scattering of obliquely incident water waves by two thin vertical barriers with gaps at different depths has been studied assuming linear theory. Using Havelock’s expansion of water wave potential, the problem is reduced to two pairs of integral equations of the first kind, one pair involving a horizontal component of velocity across the gaps and the other pair involving the difference of potentials across each wall. These two pairs of integral equations can be solved approximately by employing a Galerkin single-term approximation technique to obtain numerical estimates for the reflection and transmission coefficients. These estimates for the reflection and transmission coefficients thus obtained are seen to satisfy the energy identity. The reflection coefficient is plotted against wave number in a number of figures for different values of various parameters involved in the problem. It is observed that the reflection coefficient vanishes at discrete frequencies when the vertical barriers are identical. For nonidentical vertical barriers the reflection coefficient never vanishes, though at some wave number it becomes close to zero. The results for a single barrier and fully submerged two barriers, and for a single barrier with a narrow gap, are also recovered as special cases.

Torsion equations for lateral buckling

A torsion differential equation previously used for analysing the elastic lateral buckling of simply supported doubly symmetric beams with distributed loads acting away from the centroidal axis omits an expected term and includes an unexpected term. A different equation is derived by two different methods, either by using the calculus of variations with the second variation of the total potential, or by considering the equilibrium of the deflected and twisted beam.Four different methods are used to find solutions for the elastic buckling of beams with uniformly distributed loads. Two of these solve the differential equations numerically, either by using a computer program based on the method of finite integrals, or by making hand calculations with a single term approximation of the buckled shape. These methods produce different solutions for the two torsion differential equations.The two other methods used are based on the energy equation for lateral buckling. The first of these uses hand calculations and a limited series for the buckled shape, while the second uses a finite element computer program based on cubic deformation fields. Both of these produce solutions which agree closely with the finite integral and approximate solutions for the different differential equation derived in this paper, but are markedly different from the solutions for the previously used equation.It is concluded that the previously used torsion differential equation is in error.

Accuracy of Strain Solution of a Laminated Composite Plate via Galerkin's Method

Abstract This paper deals with the detailed analysis and discussion of nonlinear static analysis of a laminated composite plate subjected to uniform pressure load. Governing equations of the plate are derived by the use of the virtual work principle. The geometric nonlinearity effects are taken into account with the von Kármán large deflection theory of thin plates. Approximate solutions for a simply supported plate are assumed for the plate domain. The single term approximation functions are selected by considering the nonlinear static deformations of plate. The solution is obtained using the Newmark method in MATLAB software. The plate is also modeled and analyzed using ANSYS software using SHELL91 elements. The variation of membrane, linear and nonlinear bending terms are obtained along a characteristic line on the plate and the numerical and finite element results are compared. The results are discussed in detail considering the nonlinear static behavior of the laminated composite plate and the main reasons of discrepancies are discussed. Dynamic behavior of the plate is explained considering the nonlinear static behavior.

Nonlinear dynamic analysis of a laminated hybrid composite plate subjected to time-dependent external pulses

Nonlinear dynamic responses of a laminated hybrid composite plate subjected to time-dependent pulses are investigated. Dynamic equations of the plate are derived by the use of the virtual work principle. The geometric nonlinearity effects are taken into account with the von Kármán large deflection theory of thin plates. Approximate solutions for a clamped plate are assumed for the space domain. The single term approximation functions are selected by considering the nonlinear static deformation of plate obtained using the finite element method. The Galerkin Method is used to obtain the nonlinear differential equations in the time domain and a MATLAB software code is written to solve nonlinear coupled equations by using the Newmark Method. The results of approximate-numerical analysis are obtained and compared with the finite element results. Transient loading conditions considered include blast, sine, rectangular, and triangular pulses. A parametric study is conducted considering the effects of peak pressure, aspect ratio, fiber orientation and thicknesses.

Nonlinear dynamic behavior of simply supported laminated composite plates subjected to blast load

This paper deals with the analysis and discussion of nonlinear dynamic response of a laminated composite plate subjected to blast load. Dynamic equations of the plate are derived by the use of the virtual work principle. The geometric nonlinearity effects are taken into account with the von Kármán large deflection theory of thin plates. Approximate solutions for a simply supported plate are assumed for the space domain. The single term approximation functions are selected by considering the nonlinear static deformations of plate, which is obtained using finite element method. The Galerkin Method is used to obtain the nonlinear differential equations in the time domain. The finite difference method is applied to solve the system of coupled nonlinear equations. The results of approximate-numerical analysis are obtained and compared with the literature and finite element results. Good agreement is found for the character and frequencies of vibrations.

Efficient multiple-term approximations for the generalised elliptic-type integrals

AbstractThe generalised elliptic-type integral Rμ(k, α, γ)where 0 ≤ k < 1, Re(γ) > Re(α) > 0, Re(μ) ≥ −0.5, is represented in terms of the Gauss hypergeometric function by Kalla, Conde and Hubbell [8]. In 1987, Kalla, Lubner and Hubbell derived a simple-structured single-term approximation for this function in the neighbourhood of k2 = 1 in some range of the parameters α, γ and μ. Another formula which complements the parameter range was recently derived by the author. In this paper a novel technique is used in deriving multiple-term efficient approximations (in the neighbourhood of k2 = 1) which may be considered as a generalisation to the concept of the single-term approximations mentioned above. Two non-overlapping expressions which almost cover the entire range of parameters (α, γ, μ) are derived. Closed-form solutions are obtained for single- and double-term approximations (in the neighbourhood of k2 = 1). Results show that the proposed technique is superior to existing approximations for the same number of terms. Our formulation has potential application for a wide class of special functions.

Likelihood methods for MPSK modulation classification

New algorithms based on the likelihood functional (LF) and approximations thereof are proposed for the problem of classifying MPSK modulations in additive white Gaussian noise. Previously introduced classifiers for this problem are theoretically interpreted as simplified versions of the ones in here. The performance of a single-term approximation to the optimal LF classifier is evaluated analytically and is shown to be very close to that of the optimal. Furthermore, recursive algorithms for the implementation of this new quasi-log-likelihood-ratio (qLLR) classifier are derived which imply no significant increase in classifier complexity. The present method of generating classification algorithms can be generalized to arbitrary two-dimensional signal constellations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

PROBABILITY APPROXIMATIONS FOR DIVISIBLE DISCRETE DISTRIBUTIONS

SummaryIf an integer‐valued random variable can be represented as a sum of independent random variables, then powerful tools exist to derive approximations to its distribution. We apply this idea to examples in some of which it is not clear how to give a physical interpretation to the independent sum‐mands. We consider bounds on the accuracy of single term approximations, Edgeworth expansions and saddlepoint approximations for both individual probabilities and cumulative probabilities.

Analysis of Darling-Dennison Interactions and Related Phenomena in the SO 2 Molecule for the Force Field Study

The famous Darling-Dennison resonance is fully discussed for the 32S 16O 2 molecule by use of the classical treatment as well as by the Van Vleck contact transformation method. The conclusion is that the single-term approximation as a resonance phenomenon, originally proposed for the H 2O molecule by Darling and Dennison, cannot be applied to the SO 2 molecule, because the two frequencies, ν 1 and ν 3 are separated by too far a distance in the latter molecule. The force field of the SO 2 molecule was calculated up to the quartic terms in order to confirm the above conclusions.

Algorithms for the approximation of the generalized Hubbell rectangular source integrals

In this paper, we derive simply-structured, single term approximation formulae for the generalised Hubbell rectangular source integrals involving hypergeometric functions. These results are sufficiently accurate to allow a semi-quantitative assessment of the dependence of these functions on the parameters. The corresponding result for the Hubbell integral is derived as a particular case. Tables are given to compare the numerical values derived from the approximate formulae with those given earlier by Kalla et al. [Kalla S. L., Al-Saqabi B. and Conde S. (1987a) Hadronic J. 10, 221–230.].

Non-linear forced vibrations of antisymmetric rectangular cross-ply plates

A method of direct numerical integration of the frequency/time period expression is proposed to study the non-linear forced/free vibration behaviour of rectangular, antisymmetric cross-ply plates. The presence of bending-extension coupling in such plates results in a cubic non-linear term in addition to a fourth power term in the energy balance equation. The existence of this term does not allow two real and opposite roots at a maximum amplitude position unlike in isotropic, orthotropic, symmetric cross-ply, square antisymmetric cross-ply and symmetric and antisymmetric angle-ply plates, but such an energy balance equation definitely has two real roots, which are unequal in magnitude and different in sign. The extensively used perturbation method fails to predict the response of such plates, depending on the severity of the bending-extension coupling. The proposed direct integration method, even with single-term approximations for in-plane and transverse displacements, yields consistently accurate results. Response curves for several configurations of antisymmetric, simply supported rectangular cross-ply plates are presented.

Non-linear vibrations of simply supported rectangular cross-ply plates

A method of direct numerical integration of the frequency-ratio expression is proposed to study the non-linear free vibration behaviour of rectangular cross-ply laminates. The proposed method, even with single-term approximations for the admissible functions, yields results that agree very well with the existing perturbation solutions. Non-linear behaviour of the cross-ply laminates is also studied with the harmonic oscillations assumption, by using the conservation of energy and the modal equation. The results are found to be lower and upper bounds to those obtained from the direct numerical integration method. It is also observed that the arithmetic mean of the two solutions with the harmonic oscillations assumption matches very well with that of the direct numerical integration method. Non-linear vibration characteristics are obtained for several configurations of cross-ply plates. Results for orthotropic and isotropic plates are also obtained as special cases.

D9 dipole dispersion coefficients

A formula is derived for D 9, the first correction to the usual single-term approximation for the dispersion dipole of a pair of dissimilar atoms. The expression is a Casimir—Polder integral involving dipole and quadrupole polarisabilities, dipole—dipole—quadrupole and dipole— quadrupole—octopole hyperpolarisabilities on each atom. A coupled Hartree—Fock scheme for its ab initio calculation is implemented and static approximation to D 9 show it to be of the same sign as D 7 but larger in magnitude (when both are expressed in atomic units). CHF calculations on H and He give D 9(H +He −)=−3390 au, i.e. over 25 a 2 0 times D 7, and the R −9 local dipole in H…H is over 30 R −2 times the better known R −7 moment.

Analysis of the forms of the loss of stability of ribbed shells

1. An increase in the number of ribs in the stringer-reinforced shells may result in the change from one isolated form of stability loss to another in accordance with the conditions n=ks, n=ks/2, n=ks/4, etc.; at the instant of transition from one form to another, the total critical load may decrease with an increase in the number of ribs. 2. The data obtained on the basis of multiterm approximation are similar mainly to the results corresponding to single-term approximation for the even number of ribs, and also to the intermediate numbers of ribs. It is thus possible to propose a correction for the data obtained on the basis of single-term approximation for the odd number of ribs using the intermediate values of the critical loads between the adjacent even numbers of ribs. 3. The effect of the torsional stiffness of the ribs (even for Icrs/t4>Is/t4) on the critical load in the majority of cases is small and equals 22% only in a single case of all the examined cases; the magnitude of this effect will decrease even further with the reduction of torsional stiffness. 4. The critical loads determined on the basis of multiterm approximation may decrease by 18% (e. g., for 20 stringers) in comparison with the data obtained for single-term approximation.

Walsh stretch matrices and functional differential equations

This correspondence presents a new operational matrix for Stretch via Walsh functions and discusses some interesting properties of this matrix and its role in the solution of certain functional differential equations. The solutions of differential equations are obtained in a series of Walsh functions. The results are piecewise constant with minimal mean-square error. The single term approximation combined with the technique of extension [8] reduces the computational effort to a minimum.

Error Performance of Direct Sequence Spread Spectrum Systems on Non-Selective Fading Channels

The effects of fading on the performance of a spread spectrum multiple-access communication system using direct sequence modulation are investigated. General series expansions are developed for the average probability of error and specific results are given for the irreducible error rate associated with propagation over independent Rician fading channels or correlated Rayleigh fading channels. Simple single-term approximations for the irreducible error rate, which are proportional to the second moment of the cross-correlation between the spreading codes employed by the users, are derived for these cases.

Angular Distribution ofO−from Dissociative Electron Attachment toO2

The angular distribution of ${\mathrm{O}}^{\ensuremath{-}}$ produced by electron bombardment of ${\mathrm{O}}_{2}$ has been measured in the electron energy range 5.75-8.40 eV. The results show a strong energy dependence and are consistent with the theory of O'Malley and Taylor if the final $\mathrm{O}_{2}^{}{}_{}{}^{\ensuremath{-}}$ repulsive resonance state is assumed to have the symmetry $^{2}\ensuremath{\Pi}_{u}$, and if only the first two allowed partial waves of the incident electron corresponding to $L=1$ and $L=3$ contribute. The results indicate that the $L=3$ term becomes more important as energy increases and thereby demonstrate that the single-term approximation for the angular distribution does not apply for this process.

Analysis of the temperature distribution in pulled crystals

The time independent equations governing the temperature distribution in a growing crystal are solved to give both axial and radial variations of temperature. It is shown that simple single term approximations to the solutions may be used without appreciable loss of accuracy. In a given system, if a is the radius of the crystal, at the growth interface the products a 1 2 ∂T/∂Z and a ∂ 2 T/ ∂Z 2 are constant and if the crystal is longer than a critical length (about 6a 1 2 cm in germanium), the heat flow at the interface is independent of the length. The relations derived are shown to be consistent with measurements made on germanium crystals. Measurements made in the liquid show the limitations of the available theory. It is shown that crystals rotation can lead to thermal instabilities.

Nonsymmetrical Snap Buckling of Spherical Caps

Buckling of uniformly loaded clamped spherical caps is computed by energy procedure; use of single term approximations to represent initial and buckled shapes; analysis including development of boundary layer energy load as lower bound on buckling loads is examined.